Modelling, simulation and analysis of low-cost direct torque control of PMSM using hall-effect se

MODELLING, SIMULATION AND ANALYSIS OF LOW-COST DIRECT

TORQUE CONTROL OF PMSM USING HALL-EFFECT SENSORS

A Thesis

by

SALIH BARIS OZTURK

Submitted to the Office of Graduate Studies of Texas A&M University

in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE

December 2005

Major Subject: Electrical Engineering

MODELLING, SIMULATION AND ANALYSIS OF LOW-COST DIRECT

TORQUE CONTROL OF PMSM USING HALL-EFFECT SENSORS

A Thesis

by

SALIH BARIS OZTURK

Submitted to the Office of Graduate Studies of Texas A&M University

in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE

Approved by: Chair of Committee, Hamid A. Toliyat Committee Members, Prasat N. Enjeti S. P. Bhattacharyya Reza Langari Head of Department, Costas N. Georghiades

December 2005

Major Subject: Electrical Engineering

iii

ABSTRACT

Modelling, Simulation and Analysis of Low-Cost Direct Torque Control

of PMSM Using Hall-Effect Sensors. (December 2005)

Salih Baris Ozturk, B.S., Istanbul Technical University

Chair of Advisory Committee: Dr. Hamid A. Toliyat

This thesis focuses on the development of a novel Direct Torque Control (DTC)

scheme for permanent magnet (PM) synchronous motors (surface and interior types) in

the constant torque region with the help of cost-effective hall-effect sensors. This

method requires no DC-link sensing, which is a mandatory matter in the conventional

DTC drives, therefore it reduces the cost of a conventional DTC of a permanent magnet

(PM) synchronous motor and also removes common problems including; resistance

change effect, low speed and integration drift. Conventional DTC drives require at least

one DC-link voltage sensor (or two on the motor terminals) and two current sensors

because of the necessary estimation of position, speed, torque, and stator flux in the

stationary reference frame.

Unlike the conventional DTC drive, the proposed method uses the rotor reference

frame because the rotor position is provided by the three hall-effect sensors and does not

require expensive voltage sensors. Moreover, the proposed algorithm takes the

acceleration and deceleration of the motor and torque disturbances into account to

improve the speed and torque responses.

The basic theory of operation for the proposed topology is presented. A

mathematical model for the proposed DTC of the PMSM topology is developed. A

simulation program written in MATLAB/SIMULINK® is used to verify the basic

operation (performance) of the proposed topology. The mathematical model is capable

of simulating the steady-state, as well as dynamic response even under heavy load

conditions (e.g. transient load torque at ramp up). It is believed that the proposed system

iv

offers a reliable and low-cost solution for the emerging market of DTC for PMSM

drives.

Finally the proposed drive, considering the constant torque region operation, is

applied to the agitation part of a laundry washing machine (operating in constant torque

region) for speed performance comparison with the current low-cost agitation cycle

speed control technique used by washing machine companies around the world.

v

To my mother and father.

vi

ACKNOWLEDGMENTS

There is a numerous amount of people I need to thank for their advice, help,

assistance and encouragement throughout the completion of this thesis.

First of all, I would like to thank my advisor, Dr. Hamid A. Toliyat, for his

support, continuous help, patience, understanding and willingness throughout the period

of the research to which this thesis relates. Moreover, spending his precious time with

me is appreciated far more than I have words to express. I am very grateful to work with

such a knowledgeable and insightful professor. To me he is more than a professor; he is

like a father caring for his students during their education. Before pursuing graduate

education in the USA I spent a great amount of time finding a good school, and more

importantly a quality professor to work with. Even before working with Dr. Toliyat I

realized that the person who you work with is more important than the prestige at the

university in which you attend.

I would also like to thank the members of my graduate study committee,

Dr. Prasad Enjeti, Dr. S.P. Bhattacharyya, and Dr. Reza Langari for accepting my

request to be a part of the committee.

I would like to extend my gratitude to my fellow colleagues in the Advanved

Electric Machine and Power Electronics Laboratory: our lab manager Dr. Peyman Niazi,

Dr. Leila Parsa, Dr. M.S. Madani, Dr. Namhun Kim, Dr. Mesuod Haji, Dr. Sang-Shin

Kwak, Dr. Mehdi Abolhassani, Bilal Akin, Steven Campbell, Salman Talebi, Rahul

Khopkar, and Sheab Ahmed. I cherish their friendship and the good memories I have

had with them since my arrival at Texas A&M University.

Also, I would like to thank to the people who are not participants of our lab but

who are my close friends and mentors who helped, guided, assisted and advised me

during the completion of this thesis: Onur Ekinci, David Tarbell, Ms. Mina Rahimian,

Dr. Farhad Ashrafzadeh, Peyman Asadi, Sanjey Tomati, Neeraj Shidore, Steve Welch,

Burak Kelleci, Guven Yapici, Murat Yapici, Cansin Evrenosoglu, Ali Buendia, Kalyan

Vadakkeveedu, Leonardo Palma and many others whom I could not mention here.

vii

I would also like to acknowledge the Electrical Engineering department staff at

Texas A&M University: Ms. Tammy Carda, Ms. Linda Currin, Ms. Gayle Travis and

many others for providing an enjoyable and educational atmosphere.

Last but not least, I would like to thank my parents for their patience and endless

financial, and more importantly, moral support throughout my life. First, I am very

grateful to my dad for giving me the opportunity to study abroad to earn a good

education. Secondly, I am very grateful to my mother for her patience which gave me a

glimpse of how strong she is. Even though they do not show their emotion when I talk to

them, I can sense how much they miss me when I am away from them. No matter how

far away from home I am, they are always there to support and assist me. Finally, to my

parents, no words can express my gratitude for you and all the things you sacrifice for

me.

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TABLE OF CONTENTS

Page

ABSTRACT ......................................................................................................................iii

ACKNOWLEDGMENTS.................................................................................................vi

TABLE OF CONTENTS ................................................................................................viii

LIST OF FIGURES...........................................................................................................xi

LIST OF TABLES .........................................................................................................xxii

CHAPTER

I INTRODUCTION..................................................................................................1

1.1 Principles of DC Brush Motors and Their Problems .................................1 1.2 Mathematical Model of the DC Brush Motor ............................................6 1.3 AC Motors and Their Trend.....................................................................10 1.4 Thesis Outline ..........................................................................................12

II BASIC OPERATIONAL PRINCIPLES OF PERMANENT MAGNET SYNCHRONOUS AND BRUSHLESS DC MOTORS ......................................14

2.1 Permanent Magnet Synchronous Motors (PMSMs) ................................14 2.1.1 Mathematical Model of PMSM.................................................20

2.2 Introduction to Brushless DC (BLDC) Motors ........................................36 2.2.1 Mathematical Model of BLDC Motor ......................................39

2.3 Mechanical Structure of Permanent Magnet AC Motors (PMSM and BLDC .......................................................................................................45

2.4 Advantages and Disadvantages of PMSM and BLDC Motors ................49 2.5 Torque Production Comparison of PMSM and BLDC Motors ...............53 2.6 Control Principles of the BLDC Motor....................................................55 2.7 Control Principles of the PM Synchronous Motor...................................62

III MEASUREMENTS OF THE PM SPINDLE MOTOR PARAMETERS ...........67

3.1 Introduction ..............................................................................................67 3.2 Principles of Self-Commisioning with Identification ..............................68 3.3 Resistance and dq-axis Synchronous Inductance Measurements.............69 3.4 Measurment of the Back-EMF Constant and Rotor Flux Linkage ..........79 3.5 Load Angle Measurement ........................................................................81

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CHAPTER Page

IV OPEN-LOOP SIMULINK® MODEL OF THE PM SPINDLE MOTOR ...........88

4.1 Principles of PMSM Open-Loop Control ................................................88 4.2 Building Open-Loop Steady and Transient-State Models .......................90 4.3 Power Factor and Efficiency Calculations in Open-Loop

Matlab/Simulink® Model .........................................................................97

V CONVENTIONAL DIRECT TORQUE CONTROL (DTC) OPERATION OF PMSM DRIVE ...............................................................................................99

5.1 Introduction and Literature Review .........................................................99 5.2 Principles of Conventional DTC of PMSM Drive .................................107

5.2.1 Torque Control Strategy in DTC of PMSM Drive..................108 5.2.2 Flux Control Strategy in DTC of PMSM Drive......................112 5.2.3 Voltage Vector Selection in DTC of PMSM Drive ................107

5.3 Control Strategy of DTC of PMSM Drive .............................................120

VI PROPOSED DIRECT TORQUE CONTROL (DTC) OPERATION OF PMSM DRIVE ...................................................................................................124

6.1 Introduction and Principles of the Proposed DTC Drive .......................124 6.2 Position and Speed Estimation Method with Low Resolution Position

Sensor .....................................................................................................130

VII AGITATION CYCLE SPEED CONTROL OF VERTICAL-AXIS LAUNDRY MACHINE.....................................................................................135

7.1 Modern Washing Machines: Top Loaded (Vertical-axis) and Front Loaded (Horizontal-axis) Washine Machines........................................135

7.2 Introduction to Speed Control of a BLDC Spindle Motor (48 Pole/ 36 Slot) in the Agitation Cycle of a Washing Machine .........................136

7.3 Agitate Washer Controller Model in SIMPLORER® ............................138

VIII EXPERIMENTAL AND SIMULATION ANALYSIS.....................................148

8.1 Open-Loop Simulation vs. Experimental Results ..................................148 8.1.1 Open-Loop Steady-State and Transient Analysis (Simulation).............................................................................148 8.1.2 Open-Loop Steady-State and Transient Analysis (Experiment)............................................................................152 8.1.3 System Configuration for Electrical Parameters

Measurements..........................................................................154 8.1.4 The M-Test 4.0 Software ........................................................156 8.1.5 Curve Testing ..........................................................................157

8.2 Experimental vs. Simulation Results (Commercial Agitation Control).158 8.2.1 Complete Agitation Speed Control Analysis (Steady-State) ..159

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8.2.2 Complete Agitation Speed Control Analysis (Transient and Steady-States)..........................................................................169

8.2.3 Simulation Results of the Proposed DTC Method Used in Agitation Cycle Speed Control ...............................................176 8.2.3.1 Transient and Steady-State Response Under Load

Condition..................................................................181 8.2.3.2 Effects of Stator Resistance Variation in

Conventional DTC with Encoder .............................195 8.2.3.3 Drift Problem Occurs in Conventional DTC Scheme .....................................................................209 8.2.3.4 Overview of the Offset Error Occured in the

Proposed DTC Scheme ............................................213 8.2.3.5 Overview of the Rotor Flux Linkage Error Occured

in the Proposed DTC Scheme ..................................218

IX SUMMARY AND FUTURE WORK................................................................224

9.1 Summary of the Work and Conclusion ..................................................224 9.2 Future Work ...........................................................................................225

REFERENCES...............................................................................................................227

APPENDIX A ................................................................................................................232

VITA ..............................................................................................................................237

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LIST OF FIGURES

FIGURE Page

1.1. Basic model of a DC brush motor ....................................................................2 1.2. Fundamental operation of a DC brush motor (Step 1). ....................................4 1.3. Fundamental operation of a DC brush motor (Step 2). ....................................5 1.4. Fundamental operation of a DC brush motor (Step 3). ....................................6 1.5. Electrical circuit model of a separately excited

DC brush motor (transient-state)......................................................................7 1.6. Electrical circuit model of a separately excited

DC brush motor (steady-state) .........................................................................8

2.1. Two-pole three phase surface mounted PMSM. ............................................22 2.2. Q-axis leading d-axis and the rotor angle represented as ..........................26 rθ 2.3. Q-axis leading d-axis and the rotor angle represented as θ ...........................28 2.4. Line-to-line back-EMF ( ), phase back-EMFs, phase current abE

waveforms and hall-effect position sensor signals for a BLDC motor ..........38

2.5. Equivalent circuit of the BLDC drive (R, L and back-EMF).........................40 2.6. Simplified equivalent circuit of the BLDC drive

(R, (L-M) and back-EMF)..............................................................................42

2.7. Cross sectional view of four pole surface-mounted PM rotor ( d qL L= ) .......46 2.8. Cross sectional view of four pole interior-mounted PM rotor ( d qL L≠ ).......47 2.9. Ideal BLDC operation ....................................................................................55 2.10. PMSM controlled as BLDC ...........................................................................55 2.11. Voltage Source Inverter (VSI) PMAC drive..................................................55

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FIGURE Page

2.12. Voltage Source Inverter (VSI) BLDC drive ..................................................56 2.13. A typical current-controlled BLDC motor drive with position

feedback .........................................................................................................59

2.14. Six possible switching sequences for a three-phase BLDC motor.................60 2.15. PWM current regulation for BLDC motor [8] ...............................................61 2.16. Basic phasor diagram for PMSM [8] .............................................................63 2.17. Basic block diagram for high-performance torque control

for PMSM [8] .................................................................................................64 2.18. Basic block diagram for high-performance torque control

for PMSM [8] .................................................................................................65 2.19. Basic block diagram for high-performance torque control

for PMSM [8] .................................................................................................65 3.1. Vector phasor diagram for non-salient PMSM ..............................................72 3.2. Measurement of phase resistance and inductance..........................................74 3.3. Phase A voltage waveform and hall-effect signal of the same phase ............82 3.4. Rotor magnet representation along with hall-effect signals...........................83 3.5. Phase A voltage waveform and hall-effect signal with jittering effect ..........83 3.6. Experimental vs. simulated load angle (deg.) at 20 Hz (No-load) .................85 3.7. Experimental vs. simulated load angle (deg.) at 40 Hz (No-load) .................86 3.8. Experimental vs. simulated load angle (deg.) at 40 Hz (Load)......................86 3.9. Possible vector representation of PMSM .......................................................87 4.1. Basic open-loop block diagram of PMSM.....................................................89 4.2. Overall open-loop control block diagram of PMSM in SIMULINK®...........91

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FIGURE Page

4.3. Abc to stationary and stationary to dq rotor reference

frame transformation block ............................................................................91

4.4. Rotor reference frame and stationary reference frame coordinate representation ...............................................................................92

4.5. Dq rotor reference frame to stationary and stationary to

abc transformation block................................................................................93

4.6. Steady state dq-axis rotor reference frame motor model ...............................94 4.7. Rotor ref. frame, synchronous ref. frame, and stationary ref. frame

representation as vectors ................................................................................96

5.1. Eight possible voltage space vectors obtained from VSI.............................100 5.2. Circular trajectory of stator flux linkage in the stationary DQ-plane ..........101 5.3. Phasor diagram of a non-saliet pole synchronous machine in

the motoring mode .......................................................................................108

5.4. Electrical circuit diagram of a non-salient synchronous machine at constant frequency (speed) ........................................................109

5.5. Rotor and stator flux linkage space vectors

(rotor flux is lagging stator flux) ..................................................................112

5.6. Incremental stator flux linkage space vector representation in the DQ-plane ......................................................................................................112

5.7. Representation of direct and indirect components of

the stator flux linkage vector ........................................................................114

5.8. Voltage Source Inverter (VSI) connected to the R-L load...........................116 5.9. Voltage vector selection when the stator flux vector is located in sector i ..117

5.10. Basic block diagram of DTC of PMSM.......................................................120

6.1. Hall-effect sensor placement shown in DQ stationary reference frame.......125

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FIGURE Page

6.2. Representation of the rotor and stator flux linkages both in the synchronous dq and stationary DQ-planes...................................................127

6.3. Block diagram of the proposed DTC of PMSM ..........................................129

6.4. Speed and position estimation using low-resolution position sensor...........131 6.5. Representation of acceleration and speed at the sectors

and at the hall-effect sensors, respectively...................................................134

7.1. Simple flowchart of the commercial agitation speed control [57] ...............138 7.2. Reference speed profile for the agitation speed control

in a washing machine ...................................................................................139 7.3. Detailed flowchart of the simple agitation speed control

of a washing machine [58] ...........................................................................140

7.4. PWM increment during the ramp time [58] .................................................143 7.5. Speed response during ramp up when PWM is incremented

at every PT time [58] ...................................................................................144

7.6. Ideal speed profile of the agitation speed control [58].................................146 7.7. Typical practical speed response of the agitation speed control [58]...........147 8.1. Efficiency, power factor, motor current (Phase A), and

motor speed versus applied load torque (open-loop exp. and sim.) at 100 rpm with V / f ~= 2 ..........................................................................150

8.2. Dynamic speed response of exp. and sim. when V / f = 18/15 (0-37.5 rpm) .................................................................................................151

8.3. Dynamic speed response of exp. and sim. when V / f = 1 (0-37.5 rpm) .................................................................................................151

8.4. Torque response of PMSM when it is supplied by quasi-square wave

currents .........................................................................................................152

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FIGURE Page

8.5. Efficiency, power factor, motor current (Phase A), and motor speed versus applied load torque (experimental, open-loop)

at 100 rpm with V / f ~= 2 ..........................................................................154

8.6. Basic block diagram of overall system for efficiency measurements ..........155 8.7. Overall system configuration .......................................................................156 8.8. Sample curve test .........................................................................................158 8.9. Graphical representation of the sample curve test .......................................158 8.10. Input current of the single phase diode rectifier at 10 N·m of load torque..159 8.11. Input current to the single phase diode rectifier (experiment)

at 10 N·m of load torque (100 rpm).............................................................160 8.12. Voltage source obtained from the wall plug ................................................161 8.13. DC-link voltage of the VSI at 10 N·m of load torque (100 rpm) .................162 8.14. Steady-state load torque and plateau speeds (experiment and

simulation) at 10 N·m of load torque (100 rpm) ..........................................163 8.15. Steady-state electromagnetic torque (simulation) at 10 N·m of load

torque (100 rpm)...........................................................................................164

8.16. Block diagram of the overall commercialized agitation washer control with the trapezoidal voltage waveform and efficiency calculation and measurement.................................................................................................165

8.17. Experimental three-phase motor phase currents at 10 N·m of

load torque (100 rpm)...................................................................................166

8.18. Simulated three-phase motor phase currents at 10 N·m of load torque (100 rpm)...................................................................................167

8.19. Exp. PF., eff., and sim. PF., eff. vs. torque at 10 N·m (100 rpm).................167

8.20. Block diagram of the overall commercialized agitation washer control

with trapezoidal voltage waveform ..............................................................168

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FIGURE Page

8.21. Experimental and simulation speed response (transient and steady-state) at no-load......................................................................................................170

8.22. Experimental and simulation speed response (transient and steady-state)

at 5 N·m of load torque.................................................................................171

8.23. Experimental and simulation speed response (transient and steady-state) at no-load (0.7 second ramp time)................................................................172

8.24. Rotor position obtained in simulation ..........................................................173 8.25. Steady-state SIMPLORER® model of the agitation speed control ..............174 8.26. SIMPLORER® model of commercialized overall agitation speed

control system (transient+steady-state) ........................................................175

8.27. Agitation speed control test-bed...................................................................176 8.28. Load torque ( LT ), , , and at a 0.35 second ramp time emrefT emestT emactT

when the PI integrator initial value is 7 N·m................................................182 8.29. sλ

∗ (Reference Stator Flux), _s estλ and _s actλ at a 0.35 second ramp time when the PI integrator initial value is 7 N·m................................................183

8.30. sDλ and sQλ circular trajectory at a 0.35 second ramp time when

the PI integrator initial value is 7 N·m .........................................................183 8.31. Load torque ( LT ), , , and at a 0.35 second ramp time emrefT emestT emactT

when the PI integrator initial value is 8.5 N·m.............................................184

8.32. sλ∗ (Reference Stator Flux), _s estλ and _s actλ at a 0.35 second ramp time

when the PI integrator initial value is 8.5 N·m.............................................185

8.33. sDλ and sQλ circular trajectory at a 0.35 second ramp time when the PI integrator initial value is 8.5 N·m ......................................................186

8.34. Phase-A current ( ), , and when a 0.35 second ramp time asi dqracti dqresti

is used when the PI integrator initial value is 7 N·m....................................187

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FIGURE Page

8.35. Phase-A current ( ), , and when a 0.35 second ramp time asi dqracti dqresti is used when the PI integrator initial value is 8.5 N·m.................................188

8.36. Reference and actual speed, speed error (ref-act), actual and estimated

position when a 0.35 second ramp time is used when the PI integrator initial value is 7 N·m ....................................................................................189

8.37. Reference and actual speed, speed error (ref-act), and actual and

estimated position when a 0.35 second ramp time is used when the PI integrator initial value is 8.5 N·m.................................................................190

8.38. , , and at a 0.6125 second ramp time when emrefT emestT emactT

the PI integrator initial value is 7 N·m .........................................................191 8.39. sλ

∗ (reference stator flux), _s estλ and _s actλ at 0.6125 second ramp time when PI integrator initial value is 7 N·m......................................................192

8.40. sDλ (X axis) and sQλ (Y axis) circular trajectory at a 0.6125 second ramp

time when the PI integrator initial value is 7 N·m........................................193 8.41. Phase-A current ( ), , and when a 0.6125 second ramp time asi dqracti dqresti

is used when the PI integrator initial value is 7 N·m....................................194

8.42. Reference and actual speeds, speed error (ref-act), and actual and estimated positions when a 0.6125 second ramp time is used when the PI integrator initial value is 7 N·m....................................................................195

8.43. Reference and actual speeds, actual position, and three phase currents

when a 0.35 second ramp time is used (no initial torque, just ramp rate is added). Stator resistance change effect is not applied................................196

8.44. , , and at a 0.35 second ramp time. Stator resistance emrefT emestT estFlux

change effect is not applied..........................................................................197

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FIGURE Page

8.45. Load torque (TL), , , , and estimated stator flux linkage emrefT emestT emactT under step stator resistance change (1.5x sR ) at 0 second when ramp time is 0.35 second when the initial integrator value of the PI is chosen

as 7 N·m........................................................................................................198

8.46. Reference and actual speeds, actual position, and three phase currents when a 0.35 second ramp time is used (no initial torque, just ramp rate is added). Stator resistance is increased by 1.5x sR at 0 second ......................199

8.47. Reference and actual speeds, speed error (ref-act), actual position, and

three phase currents when a 0.35 second ramp time is used (no initial torque, just ramp rate is added). Stator resistance is increased by 1.5x sR at 0.675 second ............................................................................................200

8.48. , , and at a 0.35 second ramp time. Stator resistance emrefT emestT estFlux

is increased by 1.5x sR at 0.675 second .......................................................201

8.49. , , , , estimated and real rotor positions with a emestT emactT estFlux actFlux 0.35 second ramp time. Stator resistance is increased by 1.5x sR at 0.675

second...........................................................................................................202 8.50. , , , , estimated and real rotor positions with a emestT emactT estFlux actFlux

0.35 second ramp time. Stator resistance is increased by 1.5x sR at 0.35/2 second ( V).....................................................................................203 370dcV =

8.51. Reference and actual speeds, and three phase currents when a 0.35 second

ramp time is used (no initial torque, just ramp rate is added). Stator resistance is increased by 1.5x sR at 0.35/2 second ( 370dcV = V) ...............204

8.52. , , and with a 0.35 second ramp time. Stator resistance emestT emactT estFlux

is increased by 1.5x sR at 0.35/2 second ( 370dcV = V) ................................205

8.53. Estimated sDλ (X axis) and sQλ (Y axis) circular trajectory with a 0.35 second ramp time. Stator resistance is increased by 1.5x sR at 0.675 second (kp=9, ki=3)......................................................................................206

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FIGURE Page

8.54. Estimated sDλ (X axis) and sQλ (Y axis) circular trajectory with a 0.35 second ramp time. Stator resistance is increased by 1.5x sR at 0.35/2 second (kp=9, ki=3)......................................................................................207

8.55. Actual sDλ (X axis) and sQλ (Y axis) circular trajectory with a 0.35

second ramp time. Stator resistance is increased by 1.5x sR at 0.35/2 second (stopped at 0.28 s.) ...........................................................................208

8.56. Actual sDλ (X axis) and sQλ (Y axis) circular trajectory with a 0.35

second ramp time. Stator resistance is increased by 1.5x sR at 0.675 second (stopped at 0.28 s.) ...........................................................................209

8.57. Reference and actual speeds, actual position, and three phase

currents when a 0.35 second ramp time is used (no initial torque, just ramp rate is added. A 0.1 A current offset is added to the D- and Q-axes currents in the SRF ( 370dcV = V).................................................................211

8.58. Load torque (TL), , , , and actual stator flux linkage. emrefT emactT estFlux

A 0.1 A offset occurs on the D- and Q-axes currents in the SRF when the ramp time is 0.35 second..............................................................................212

8.59. Estimated sDλ (X axis) and sQλ (Y axis) circular trajectory with a 0.35

second ramp time. A 0.1 A offset is included in the D- and Q-axes currents in the SRF ( 370dcV = V). (kp=9, ki=3) ..........................................213

8.60. Reference and actual speeds, speed error (ref-act), estimated and

actual positions when a 0.35 second ramp time is used (7.5 N·m. initial torque is used in the PI’s integrator with appropriate ramp rate). A 0.1 A offset is included in the d- and q-axes currents in the RRF ( V) ...214 370dcV =

8.61. Phase-A current ( ), , and when a 0.35 second ramp time is asi dqracti dqresti

used with a PI integrator initial value of 7.5 N·m. A 0.1 A offset is include to the d- and q-axes currents in the RRF ( 370dcV = V)...................215

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FIGURE Page

8.62. Load torque (TL), , , and . A 0.1 A offset occurs on d- and q-axes currents in the RRF when ramp time is 0.35 second ( 3 V)................................................................................................216

emrefT emestT emactT

70dcV =

8.63. , , when a 0.35 second ramp time is used with a PI integrator initial value of 7.5 N·m. A 0.1 A offset is included in the d- and q-axes currents in the RRF (

refFlux estFlux actualFlux

370dcV = V)..........................................217 8.64. Estimated sDλ (X axis) and sQλ (Y axis) circular trajectory at a 0.35

second ramp time with a PI integrator initial value of 7.5 N·m. A 0.1 A offset is included in the d- and q-axes currents in the RRF ( V). (kp=9, ki=3)..................................................................................................218

370dcV =

8.65. Reference and actual speeds, speed error (ref-act), estimated and

actual positions when a 0.35 second ramp time is used (7 N·m. initial torque is used in the PI’s integrator with appropriate ramp rate). 0.7x is used at 0.35/2 sec. in the motor model (

rλ370dcV = V). (kp=9, ki=3) ..........219

8.66. Phase-A current ( ), , and when a 0.35 second ramp time is

used with a PI integrator initial value of 7 N·m. 0.7x is used at 0.35/2 second in the motor model (

asi dqracti dqresti

rλ370dcV = V). (kp=9, ki=3)...............................220

8.67. Load torque (TL), , , and . 0.7x is used at emrefT emestT emactT rλ

0.35/2 second in the motor model ( 370dcV = V). (kp=9, ki=3)....................221

8.68. refFlux , , when a 0.35 second ramp time is used with a PI integrator initial value of 7 N·m. 0.7x is used at 0.35/2 second in the motor model ( V). (kp=9, ki=3).....................................................222

estFlux actualFlux

rλ370dcV =

8.69. Estimated sDλ (X axis) and sQλ (Y axis) circular trajectory at a 0.35

second ramp time with a PI integrator initial value of 7 N·m. 0.7x is used 0.35/2 second in the motor model (

rλ370dcV = V). (kp=9, ki=3) ...........223

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FIGURE Page

A-1. Specifications of the Fujitsu MB907462A microcontroller.. .......................234

A-2. Theoretical background of active power, apparent power, power factor and connection of the power analyzer to the load (motor)...........................236

A-3. MATLAB/SIMULINK® model of the proposed DTC ................................237

xxii

LIST OF TABLES

TABLE Page

3.1 Measurements of Self-Resistance and Self-Inductance .................................76

3.2 Measurements of Line-to-Neutral Back-EMF for Each Phase (0-50 rpm) ...................................................................................81

5.1 Switching Table............................................................................................118

1

CHAPTER I

INTRODUCTION

1.1. Principles of DC Brush Motors and Their Problems

DC motors have been the most widespread choice for use in high performance

systems. The main reason for their popularity is the ability to control their torque and

flux easily and independently. In DC brush machines, the field excitation that provides

the magnetizing current is occasionally provided by an external source, in which case the

machine is said to be separately excited. In particular, the separately excited DC motor

has been used mainly for applications where there was a requirement for fast response

and four-quadrant operation with high performance near zero speed.

Generally in DC brush motors, flux is controlled by manipulating field winding

current and torque by changing the armature winding current. The trade-off is less

rugged motor construction, which requires frequent maintenance and an eventual

replacement of the brushes and commutators. It also precludes the use of a DC motor in

hazardous environments where sparking is not permitted. Moreover, there is a potential

drop called ‘contact potential difference’, associated with this arrangement, and is

usually in the range of 1-1.5 V, leading to a drop in the effective input voltage.

The well-known DC brush motor, like any other rotating machine, has a stator

and rotor (as shown in Fig. 1.1). On the stator (stationary part), there is a magnetic field

which can be provided either by permanent magnets or by excited field windings on the

stator poles. On the rotor, the main components are the armature winding, armature core,

a mechanical switch called commutator which rotates, and a rotor shaft. The commutator

___________________ This thesis follows the style and format of IEEE Transactions on Industry Applications.

2

segments are insulated from one another and from the clamp holding them. In addition to

that brushes, the stationary external components of the rotor, together with the

commutator act not only as rotary contacts between the coils of the rotating armature and

the stationary external circuit, but also as a switch to commutate the current to the

external DC circuit so that it remains unidirectional even though the individual coil

voltages are alternating.

u

vw

+

-

dcV

N

S

Fig. 1.1. Basic model of a DC brush motor.

The maximum torque is produced when the magnetic field of the stator and the

rotor are perpendicular to each other (can be seen in Figs. 1.2 through 1.4,

hypothetically). The commutator makes it possible for the rotor and stator magnetic

fields to always be perpendicular. The commutation thus plays a very important part in

the operation of the DC brush motor. It causes the current through the loop to reverse at

the instant when unlike poles are facing each other. This causes a reversal in the polarity

of the field, changing attractive magnetic force into a repulsive one causing the loop to

continue to rotate.

3

When applying a voltage at the brushes, current flows through two of the coils.

This current interacts with the magnetic field of the permanent magnet and produces

torque. This torque causes to move. When the motor moves the brushes will switch to a

different coil automatically causing the rotor to turn further. If the voltage (armature) is

increased it will turn faster and if the magnetic field of permanent magnet is higher then

it will produce more torque.

Since the back-EMF generated in the coil is short-circuited by the brush, a large

current flows causing sparking at the interface of the commutator and the brushes, as

well as causing heating and the production of braking torque. In order to minimize this

problem, commutation is carried out in the magnetic field crossover region. Even after

taking these measures, because of the distortion of the effective magnetic flux due to the

armature reaction, some back-EMF is still generated in the coils in the magnetic filed

crossover region. It is desirable to minimize the crossover region in order to maximize

the utilization of the motor [1].

In general DC motors, the applied voltage (EMF) is never going to be greater

than back-EMF. The difference between the applied EMF (voltage) and back-EMF is

always such that current can flow in the conductor and produce motion.

Fundamental operation of the DC motor is explained from Figs. 1.2 through 1.4

as follows:

The direction of current flow from the DC voltage source in the figures is based

on electron theory in which current flows from the negative terminal of a source of

electricity to the positive terminal. On the contrary, the older convention supposes that

current flows from positive terminal of a source of electricity through to the negative

terminal [2].

The coil’s pole pair positioning, shown in the figures, is decided by using

Fleming’s left hand rule for generators. If a coil resides in a magnetic field and the

current and rotation of the coil are known, then direction of the magnetic field for the

coil can be found easily by using Fleming’s left hand rule. This rule states that if the

thumb and the first and middle fingers of the left hand are perpendicular to one another,

4

with the first and middle fingers pointing in the flux direction and the thumb pointing in

the direction of motion of the conductor, the middle finger will point in the direction in

which the current flows [2].

With the loop in Fig. 1.2, the current flowing through the coil makes the top of

the loop a north pole and the underside a south pole. This is found by applying the left-

hand rule under the assumption of the back-EMF (direction) is opposite of the direction

of current flow which is provided by DC voltage source [2].

N

S

NS

Fig. 1.2. Fundamental operation of a DC brush motor (Step 1).

The magnetic poles of the loop will be repelled by the like poles and attracted by

the corresponding opposite poles of the field. The coil will therefore rotate clockwise,

attempting to bring the unlike poles together.

When the loop has rotated through 90 degrees, shown in Fig. 1.3, commutation

takes place, and the current through the loop reverses its direction. As a result, the

magnetic field generated by the loop is reversed. Now, like poles face each other which

means that they repel each other and the loop continues to rotate in an attempt to bring

unlike poles together [2].

5

S

N

NS


Fig. 1.4 shows the loop position after being rotated 180 degrees from Fig. 1.3.

Now the situation is the same as when the loop was back in the position shown in

Fig. 1.3. Commutation takes place once again, and the loop continues to rotate. In this

very basic DC brush motor example, two commutator segments used with one coil loop

for simplicity. Having a small number of commutator segments in DC brush motor

causes torque ripples. As the number of segments increases, the torque fluctuation

produced by commutation is greatly reduced. In a practical machine, for example, one

might have as many as 60 segments, and the variation of the load angle between stator

magnetic flux and rotor flux would only vary ± degrees, with a fluctuation of less than

1 percent. Thus, the DC brush motor can produce a nearly constant torque [3].

3

6

N

S

NS


1.2. Mathematical Model of the DC Brush Motor

Fig. 1.5 depicts the electrical circuit model of a separately excited DC motor. The

field excitation is shown as a voltage, fV , which generates a field current, fI , that flows

through a variable resistor (which permits adjustment of the field excitation,) fR , and

through the field coil, fL . The armature circuit, on the other hand, consists of a

back-emf, , an armature resistance, bE aR , and an armature voltage, V . a

7

fI

fL

fR

fV

aL

aR

bE

aV+ +

+

-

- -

aI

T

LTmω

Fieldcircuit

Armaturecircuit

Fig. 1.5. Electrical circuit model of a separately excited DC brush motor

(transient-state).

In the motor mode, > , armature current, aV bE aI , flows into the machine. Thus,

according to the circuit model of Fig. 1.6, the operation of a DC brush motor at

steady-state, considering the inductance terms as zero, is described by the following

equation:

a a aV R I E= + b (armature circuit at steady-state) (1.1)

ff

f

VI

R= (field circuit at steady-state) (1.2)

8

fI

fR

fV

aR

bE

aV+ +

+

-

--

aI

T

LTmω

Fieldcircuit

Armaturecircuit

Mechanical Load

Fig. 1.6. Electrical circuit model of a separately excited DC brush motor (steady-state).

The equations describing the dynamic behavior of a separately excited DC brush

motor, as its circuit model in Fig. 1.5 depicts, are as follows:

( )( ) ( ) ( )aa a a a b

dI tV t R I t L E tdt

= + + (Armature circuit at transient-state) (1.3)

( )( ) ( ) f

f f f f

dI tV t R I t L

dt= + (Field circuit at transient-state) (1.4)

The torque developed by the motor can be written as follows:

( )( ) ( ) mem L m

d tT t T B t Jdtω

ω= + + (1.5)

In that equation, the total moment of inertia is represented by such that the

motor is assumed to be rigidly connected to an inertial load. Friction losses in the load

are represented by a viscous friction coefficient by

J

B , and load torque is represented by

, which is typically either constant or some function of speed, . Using these LT mω

9

conventions, the electromechanical (developed) torque of the DC brush motor can be

written as follows:

Since the electromechanical torque is related to the armature and field currents

by (1.6), (1.5) and (1.6) they are coupled to each other. This coupling may be expressed

as follows:

( ) ( )em T aT t k I tφ= (1.6)

where is called the torque constant and is related to the geometry and magnetic

properties of the structure.

Tk

Rotation of the armature conductors in the field generated by field excitation

causes a back-EMF, , in a direction that opposes the rotation of the armature. This

back-EMF is given by the expression:

bE

b aE k φω= m

aIφ

a

(1.7)

where is called the armature constant which is related to the geometry and magnetic

properties of the structure (like does). The constants and in (1.6) and (1.7) are

related to geometry factors, such as the dimension of the rotor, the number of turns in the

armature winding, and the properties of materials, such as the permeability of the

magnetic materials.

ak

Tk ak Tk

The mechanical generated output power is given by: mP

m m em m TP T kω ω= = (1.8)

where is the mechanical speed in rad/s. mω

The electric power dissipated by the motor is given by the product of the

back-EMF and the armature current which is shown as follows:

e bP E I= (1.9)

The ideal energy-conversion case, , will equal . m eP P= Tk ak

If the angular speed is denoted as rad/s, then can be expressed as: ak

10

2apNk

Mπ= (1.10)

where is the number of magnetic poles, is the number of conductors per coil and p N

M is the number of parallel paths in armature winding.

Furthermore, (1.6) can be substituted into (1.5) which yields the following

equation:

( )( ) ( ) mT a L m

d tk I t T b t Jdtω

φ ω= + + (1.11)

In the case of separately excited DC brush motors as shown in Fig. 1.6, the field

flux is established by a separate field excitation, therefore the flux equation can be

written as follows:

ff f f

NI k I

Rφ= = (1.12)

where fN is the number of turns in the field coil, R is the reluctance of the field circuit

and fI is the field current.

1.3. AC Motors and Their Trends

Unlike DC brush motors, AC motors such as Permanent Magnet AC motors

(PMSM, and BLDC motors), and Induction Motors (IM) are more rugged meaning that

they have lower weight and inertia than DC motors. The main advantage of AC motors

over DC motors is that they do not require an electrical connection between the

stationary and rotating parts of the motor. Therefore, they do not need any mechanical

commutator and brush, leading to the fact that they are maintenance free motors. They

also have higher efficiency than DC motors and a high overload capability.

11

All of the advantages listed above label AC motors as being more robust, quite

cheaper, and less prone to failure at high speeds. Furthermore, they can work in

explosive or corrosive environments because they don’t produce sparks.

All the advantages outlined above show that AC motors are the perfect choice for

electrical to mechanical conversion. Usually mechanical energy is required not at a

constant speeds but variable speeds. Variable speed control for AC drives is not a trivial

matter. The only way of producing variable speeds AC drives is by supplying the motor

with a variable amplitude and frequency three phase source.

Variable frequency changes the motor speed because the rotor speed depends on

the speed of the stator magnetic field which rotates at the same frequency of the applied

voltage. For example, the higher the frequency of the applied voltage the higher the

speed. A variable voltage is required because as the motor impedance reduces at low

frequencies the current has to be limited by means of reducing the supply voltage [4].

Before the days of power electronics and advanced control techniques, such as

vector control and direct torque control AC motors have traditionally been unsuitable for

variable speed applications. This is due to the torque and flux within the motor being

coupled, which means that any change in one will affect the other.

In the early times, very limited speed control of induction motors was achieved

by switching the three-stator windings from delta connection to star connection,

allowing the voltage at the motor windings to be reduced. If a motor has more than three

stator windings, then pole changing is possible, but only allows for certain discrete

speeds. Moreover, a motor with several stator windings is more expensive than a

conventional three phase motor. This speed control method is costly and inefficient [5].

Another alternative way of speed control is achieved by using wound rotor

induction motor, where the rotor winding ends are connected to slip rings. This type of

motor however, negates the natural advantages of conventional induction motors and it

also introduces additional losses by connecting some impedance in series with the stator

windings of the induction motor. This results in very poor performance [5].

12

At the time the above mentioned methods were being used for induction motor

speed control, DC brush motors were already being used for adjustable speed drives with

good speed and torque performance [5].

The goal was to achieve an adjustable speed drive with good speed

characteristics compared to the DC brush motor. Even after discovering of the AC

asynchronous motor, also named induction motor, in 1883 by Tesla, more than six

decades later of invention of DC brush motors, capability of adjustable speed drives for

induction motors is not as easy as DC brush motors.

Speed control for DC motors is easy to achieve. The speed is controlled by

applied voltage; e.g. the higher the voltage the higher the speed. Torque is controlled by

armature current; e.g. the higher the current the higher the torque. In addition, DC brush

motor drives are not only permitted four quadrant operations but also provided with wide

power ranges.

Recent advances in the development of fast semiconductor switches and cost-

effective DSPs and micro-processors have opened a new era for the adjustable speed

drive. These developments have helped the field of motor drives by shifting complicated

hardware control structures onto software based advanced control algorithms. The result

is a considerable improvement in cost while providing better performance of the overall

drive system. The emergence of effective control techniques such as vector and direct

torque control, via DSPs and microprocessors allow independent control of torque and

flux in an AC motor, resulting in achievement of linear torque characteristics resembling

those of DC motors.

1.4. Thesis Outline

This thesis is mainly organized as follows: First, the modeling and analysis of the

simple low-cost BLDC motor speed control on a laundry washing machine during the

agitation part of a washing cycle using Ansoft\SIMPLORER® is presented and compared

13

with the experimental ones for verification of the simulation model. To accomplish this,

motor parameters are measured to be used in the simulation platforms without

considering the saturation and temperature effects on the measured parameters.

Second, open-loop, steady-state and transient MATLAB/SIMULINK® models

are built in the dq-axis rotor reference frame. Also, to verify the open-loop simulation

models, open-loop, steady-state and transient test-beds are built. Open-loop experimental

tests are conducted under no-load and load conditions using a three phase AC power

source and results are discussed.

Third, the theoretical basis behind the simple low-cost speed control of a BLDC

motor is explained and the steady-state simulation model is built in SIMPLORER® for

comparison with the experimental results.

Fourth, overall transient and steady-state speed control of the agitation cycle in a

laundry washing machine is modeled in SIMPLORER® and verified with the

experimental tests under no-load and load conditions.

As a final goal, the proposed direct torque control of the PMSM method is

developed in MATLAB/SIMULINK® using the rotor reference frame with the help of

cheap hall-effect sensors without using any DC-link voltage sensing. This method is then

applied to the agitation cycle of washing machine speed control system for the speed

performance comparison of both methods.

The suggested method over current techniques is tested under heavy transient

load conditions resembling real washing machine load characteristics. Possible

disadvantages that may be observed in the proposed method are also considered, such as

offset and the rotor flux linkage amplitude change effect which occurs when temperature

increases on the machine. Advantages of the proposed speed control over current

methods are discussed and the results are compared.

14

CHAPTER II

BASIC OPERATIONAL PRINCIPLES OF PERMANENT MAGNET

SYNCHRONOUS AND BRUSHLESS DC MOTORS

2.1. Permanent Magnet Synchronous Motors (PMSMs)

Recent availability of high energy-density permanent magnet (PM) materials at

competitive prices, continuing breakthroughs and reduction in cost of powerful fast

digital signal processors (DSPs) and micro-controllers combined with the remarkable

advances in semiconductor switches and modern control technologies have opened up

new possibilities for permanent magnet brushless motor drives in order to meet

competitive worldwide market demands [1].

The popularity of PMSMs comes from their desirable features [4]:

• High efficiency

• High torque to inertia ratio

• High torque to inertia ratio

• High torque to volume ratio

• High air gap flux density

• High power factor

• High acceleration and deceleration rates

• Lower maintenance cost

• Simplicity and ruggedness

• Compact structure

• Linear response in the effective input voltage

15

However, the higher initial cost, operating temperature limitations, and danger of

demagnetization mainly due to the presence of permanent magnets can be restrictive for

some applications.

In permanent magnet (PM) synchronous motors, permanent magnets are

mounted inside or outside of the rotor. Unlike DC brush motors, every brushless DC (so

called BLDC) and permanent magnet synchronous motor requires a “drive” to supply

commutated current. This is obtained by pulse width modulation of the DC bus using a

DC-to-AC inverter attached to the motor windings. The windings must be synchronized

with the rotor position by using position sensors or through sensorless position

estimation techniques. By energizing specific windings in the stator, based on the

position of the rotor, a rotating magnetic field is generated. In permanent magnet ac

motors with sinusoidal current excitation (so called PMSM), all the phases of the stator

windings carry current at any instant, but in permanent magnet AC motors with

quasi-square wave current excitation (BLDC), which will be discussed in more detail

later, only two of the three stator windings are energized in each commutation sequence

[1].

In both motors, currents are switched in a predetermined sequence and hence the

permanent magnets that provide a constant magnetic field on the rotor follow the

rotating stator magnetic field at a constant speed. This speed is dependent on the applied

frequency and pole number of the motor. Since the switching frequency is derived from

the rotor, the motor cannot lose its synchronism. The current is always switched before

the permanent magnets catch up, therefore the speed of the motor is directly proportional

to the current switching rate [5].

Recent developments in the area of semiconductor switches and cost-effective

DSPs and micro-processors have opened a new era for the adjustable speed motor

drives. Such advances in the motor related sub-areas have helped the field of motor

drives by replacing complicated hardware structures with software based control

algorithms. The result is considerable improvement in cost while providing better

performance of the overall drive system [6].

16

Vector control techniques, including direct torque control incorporating fast

DSPs and micro-processors, have made the application of induction motor, synchronous

motor, recently developed PMSM, and BLDC motor drives possible for high

performance applications where traditionally only DC brush motor drives were applied.

In the past, such control techniques would have not been possible because of

complex hardware and software requirement to solve the sophisticated algorithms.

However with the recent advances in the field of power electronics, microprocessor, and

DSPs this phenomenon is solved.

Like DC brush motors, torque control in AC motors is achieved by controlling

the motor currents. However, unlike DC brush motor, in AC motors both the phase angle

and the modulus of the current has to be controlled, in other words the current vector

should be controlled. That is why the term “vector control” is used for AC motors. In

DC brush motors the field flux and armature MMF are coupled, whereas in AC motors

the rotor flux and the spatial angle of the armature MMF requires external control.

Without this type of control, the spatial angle between various fields in the AC motor

will vary with the load and cause unwanted oscillating dynamic response. With vector

control of AC motors, the torque and flux producing current components are decoupled

resembling a DC brush motor, the transient response characteristics are similar to those

of a separately excited DC motor, and the system will adapt to any load disturbances

and/or reference value variations as fast as a brushed-DC motor [4].

The primitive version of the PMAC motor is the wound-rotor synchronous motor

which has an electrically excited rotor winding which carries a DC current providing a

constant rotor flux. It also usually has a three-phase stator winding which is similar to

the stator of induction motors.

The synchronous motor is a constant-speed motor which always rotates at

synchronous speed depending on the frequency of the supply voltage and the number of

poles. The permanent magnet synchronous motor is a kind of synchronous motor if its

electrically excited field windings are replaced by permanent magnets which provide a

constant rotor magnetic field.

17

The main advantages of using permanent magnets over field excitation circuit

used by conventional synchronous motors are given below [4]:

• Elimination of slip-rings and extra DC voltage supply.

• No rotor copper losses generated in the field windings of wound-field

synchronous motor.

• Higher efficiency because of fewer losses.

• Since there is no circuit creating heat on the rotor, cooling of the motor

just through the stator in which the copper and iron loses are observed is

more easily achieved.

• Reduction of machine size because of high efficiency.

• Different size and different arrangements of permanent magnets on the

rotor will lead to have wide variety of machine characteristics.

PMAC motors have gained more popularity especially after the advent of high

performance rare-earth permanent magnets, like samarium cobalt and neodymium-boron

iron which surpass the conventional magnetic material in DC brush and induction

motors and are becoming more and more attractive for industrial applications. The positive specific characteristics of PMAC motors explained above make

them highly attractive candidates for several classes of drive applications, such as in

servo-drives containing motors with a low to mid power range, robotic applications,

motion control system, aerospace actuators, low integral-hp industrial drives, fiber

spinning and so on. Also high power rating PMAC motors have been built, for example,

for ship propulsion drives up to 1 MW. Recently two major sectors of consumer market

are starting to pay more attention to the PM motor drive due to its features [7].

PMAC motors have many advantages over DC brush motors and induction

motors. Most of these are summarized as [1]:

• High dynamic response

• High efficiency providing reduction in machine size

• Long operating life

• Noiseless operation

18

• High power factor

• High power to weight ratio; considered the best comparing to other

available electric motors

• High torque to inertia ratio; providing quick acceleration and deceleration

for short time

• High torque to volume ratio

• High air-gap flux density

• Higher speed ranges

• Better speed versus torque characteristics

• Lower maintenance cost

• Simplicity and ruggedness

• Compact design

• Linear response

• Controlled torque at zero speed

Compared to IMs, PMSMs have some advantages, such as higher efficiency in

steady-state, and operate constantly at synchronous speed. They do not have losses due

to the slip which occurs when the rotor rotates at a slightly slower speed than the stator

because the process of electromagnetic induction requires relative motion called “slip”

between the rotor conductors and the stator rotating field that is special to IM operation.

The slip makes the induction motor asynchronous, meaning that the rotor speed is no

longer exactly proportional to the supply frequency [1].

In IMs, stator current has both magnetizing and torque-producing currents. On

the other hand, in PMSMs there is no need to supply magnetizing current through the

stator for constant air-gap flux. Since the permanent magnet generates constant flux in

the rotor, only the stator current is needed for torque production. For the same output,

the PMSM will operate at a higher power factor and will be more efficient than the IM

[1].

Finally, since the magnetizing is provided from the rotor circuit by permanent

magnets instead of the stator, the motor can be built with a large air gap without losing

19

performance. According to the above results, the PMSM has a higher efficiency, torque

to ampere rating, effective power factor and power density when compared with an IM.

Combining these factors, small PM synchronous motors are suitable for certain high

performance applications such as robotics and aerospace actuators in which there is a

need for high torque/inertia ratio [1].

Induction machine drives have been widely used in industry. Squirrel-cage

induction machines have become popular drive motors due to their simple, rugged

structure, ease of maintenance, and low cost. In pumps, fans, and general-purpose drive

systems, they have been a major choice. In servo applications, permanent magnet (PM)

machine drives are more popular due to their high output torque performance. They are

also actively used in high-speed applications, not only as motors/starters but also as

generators, as in the flywheel energy storage system (FEES), electric vehicle (EV), and

industrial turbo-generator system (ITG) [7].

The smaller the motor, the more preferable it is to use permanent magnet

synchronous motors. As motor size increases the cost of the magnetics increases as well

making large PMACs cost ineffective. There is no actual “breakpoint” under which

PMSMs outperform induction motors, but the 1-10 kW range is a good estimate of

where they do [8].

As compared with induction and conventional wound-rotor synchronous motors

(WRSM), PMSMs have certain advantages over both motors, including the fact that

there is no field winding on the rotor as compared to the WRSM, therefore there are no

attendant copper losses. Moreover, there is no circuit in the rotor but only in the stator

where heat can be removed more easily [1].

Compared with conventional WRSMs, elimination of the field coil, DC supply,

and slip rings result in a much simpler motor. In a PMSM, there is no field excitation

control. The control is provided by just stator excitation control. Field weakening is

possible by applying a negative direct axis current in the stator to oppose the rotor

magnet flux [1].

20

Elimination of the need for separate field excitation results in smaller overall size

such that for the same field strength the diameter of a PMSM is considerably smaller

than its wound field counterpart, providing substantial savings in both size and weight.

2.1.1. Mathematical Model of PMSM

In a permanent magnet synchronous motor (PMSM) where the inductances vary

as a function of the rotor angle, the two-phase (d-q) equivalent circuit model is a perfect

solution to analyze the multiphase machines because of its simplicity and intuition.

Conventionally, a two-phase equivalent circuit model instead of complex three-phase

model has been used to analyze reluctance synchronous machines [9]. This theory is

now applied in the analysis of other types of motors including PM synchronous motors,

induction motors etc.

In this section, an equivalent two-phase circuit model of a three-phase PM

synchronous machine (interior and surface mount) is derived in order to clarify the

concept of the transformation (Park) and the relation between three-phase quantities and

their equivalent two-phase quantities.

Throughout the derivation of the two-phase (d-q) mathematical model of PMSM,

the following assumptions are made [6]:

• Stator windings produce sinusoidal MMF distribution. Space harmonics

in the air-gap are neglected.

• Air-gap reluctance has a constant component as well as a sinusoidal

varying component.

• Balanced three-phase sinusoidal supply voltage is considered.

• Eddy current and hysteresis effects are neglected.

• Presence of damper windings is not considered; PM synchronous motors

used today rarely have that kind of configuration.

Comparing a primitive version of a PMSM with wound-rotor synchronous motor,

the stator of a PMSM has windings similar to those of the conventional wound-rotor

21

synchronous motor which is generally three-phase, Y-connected, and sinusoidally

distributed. However, on the rotor side instead of the electrical-circuit seen in the

wound-rotor synchronous motor, constant rotor flux ( ) provided by the permanent

magnet in/on the rotor should be considered in the d-q model of a PMSM.

rλ

The space vector form of the stator voltage equation in the stationary reference

frame is given as:

ss s s

dv r idtλ

= + Equation Section 2(2.1)

where, sr , sv , si , and sλ are the resistance of the stator winding, complex space vectors

of the three phase stator voltages, currents, and flux linkages, all expressed in the

stationary reference frame fixed to the stator, respectively. They are defined as:

2

2

2

2 [ ( ) ( ) ( )]32 [ ( ) ( ) ( )]32 [ ( ) ( ) ( )3

s sa sb sc

s sa sb sc

s sa sb sc

v v t av t a v t

i i t ai t a i t

t a t a tλ λ λ λ

= + +

= + +

= + + ]

(2.2)

The resultant voltage, current, and flux linkage space vectors shown in (2.2) for

the stator are calculated by multiplying instantaneous phase values by the stator winding

orientations in which the stator reference axis for the a-phase is chosen to the direction

of maximum MMF. Reference axes for the b- and c- stator frames are chosen 120 and

(electrical degree) ahead of the a-axis, respectively. Fig. 2.1 illustrates a

conceptual cross-sectional view of a three-phase, two-pole surface PM synchronous

motor along with the two-phase d-q rotating reference frame.

240

22

as

as′

θcs

bs′ cs′

bs

q axis d axis

a axis

Fig. 2.1. Two-pole three phase surface mounted PMSM.

Symbols used in (2.2) are explained in detail below:

a , and are spatial operators for orientation of the stator windings; ,

.

2a 2 /3ja e π=2 4ja e π= / 3

sav , sbv , and scv are the values of stator instantaneous phase voltages.

sai , sbi , and sci are the values of stator instantaneous phase currents.

saλ , sbλ , and scλ are the stator flux linkages and are given by:

sa aa a ab b ac c ra

sb ab a bb b bc c rb

sc ac a bc b cc c r

L i L i L iL i L i L iL i L i L i

λ λλλ λ

= + + +

= + + +

= + + + c

λ (2.3)

where

aaL , , and are the self-inductances of the stator a-phase, b-phase, and

c-phase respectively.

bbL ccL

23

abL , , and are the mutual inductances between the a- and b-phases, b- and

c-phases, and a- and c-phases, respectively.

bcL acL

raλ , , and are the flux linkages that change depending on the rotor angle

established in the stator a, b, and c phase windings, respectively, due to the presence of

the permanent magnet on the rotor. They are expressed as:

rbλ rcλ

cos

cos( 120 )

cos( 120 )

ra r

rb r

rc r

λ λ θ

λ λ θ

λ λ θ

=

= −

= +

(2.4)

In (2.4), represents the peak flux linkage due to the permanent magnet. It is

often referred to as the back-EMF constant, .

rλ

ek

It can be seen in Fig. 2.1 that the direction of permanent magnet flux is chosen as

the d-axis, while the q-axis is ahead of the d-axis. The symbol, θ , represents the

angle of the q-axis with respect to the a-axis.

90

Note that in the flux linkage equations, inductances are the functions of the rotor

angle, . Self-inductance of the stator a-phase winding, , including leakage

inductance and a- and b-phase mutual inductance, , have the form:

θ aaL

ab baL L=

0

0

cos(2 )1 cos(2 )2 3

aa ls ms

ab ba ms

L L L L

L L L L

θ

πθ

= + −

= =− − −2 (2.5)

where

lsL is the leakage inductance of the stator winding due to the armature leakage

flux.

0L is the average inductance; due to the space fundamental air-gap flux;

01 ( )2 q dL L L= + ,

24

msL is the inductance fluctuation (saliency); due to the rotor position dependent

on flux; 1 ( )2ms d qL L L= − .

For mutual inductance, , in the above equation, a -(1/2) coefficient

appears due to the fact that stator phases are displaced by 120 , and .

ab baL L=

cos(120 ) 1/ 2=−

Similar to that of , but with θ replaced by aaL 2( )3π

θ− and 4(3π

θ− ) , b-phase

and c-phase self-inductances, and can also be obtained, respectively. bbL ccL

Also, a similar expression for and can be provided by replacing θ in

(2.5) with

bcL acL

2( )3π

θ− and 4(3π

θ− ) , respectively.

All stator inductances are represented in matrix form below:

0 0 0

0 0 0

0 0 0

1 1cos 2 cos 2( ) cos 2( )2 3 2

1 2 1cos 2( ) cos 2( ) cos 2( )2 3 3 21 1cos 2( ) cos 2( ) cos 2( )2 3 2

ls ms ms ms

ss ms ls ms ms

ms ms ls ms

L L L L L L L

L L L L L L L L

L L L L L L L

π πθ θ

π πθ θ

π πθ θ π

⎡ ⎤⎢ ⎥+ − − − − − − +⎢ ⎥⎢ ⎥⎢ ⎥= − − − + − − − − −⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥− − + − − + + − +⎢ ⎥⎢ ⎥⎣ ⎦

3

23

θ

θ π

θ

(2.6)

It is evident from (2.6) that the elements of ssL are a function of the rotor angle

which varies with time at the rate of the speed of rotation of the rotor [10].

Under a three-phase balanced system with no rotor damping circuit and knowing

the flux linkages, stator currents, and resistances of the motor, the electrical three-phase

dynamic equation in terms of phase variables can be arranged in matrix form similar to

that of (2.1) written as:

[ ] [ ][ ] [ ]s s s sdv r idt

= + Λ (2.7)

where,

25

[ ] [ ]

[ ] [ ]

[ ] [ ]

[ ] [ ]

, ,

ts sa sb sc

ts sa sb sc

ts a b c

ts sa sb sc

v v v v

i i i i

r diag r r r

λ λ λ

=

=

=

Λ =

(2.8)

In (2.7), [ ]sv , [ ]si , and [ ]sΛ refer to the three-phase applied voltages, three-phase

stator currents, and three-phase stator flux linkages in matrix forms as shown in (2.8),

respectively. Furthermore, [ ]sr is the diagonal matrix in which under balanced

three-phase conditions, all phase resistances are equal to each other and represented as a

constant, = = s a br r r r= c

]

, not in matrix form.

The matrix representation of the flux linkages of the three-phase stator windings

can also be expressed with using equations (2.4), (2.6) and (2.8) as

[ ] [ ] [s ss s rL iΛ = + Λ (2.9)

where, ssL is the stator inductance matrix varying with rotor angle.

[ ]rΛ is the flux linkage matrix due to permanent magnet; [ ] [ ] tr ra rb rλ λ λΛ = c

As it was discussed before, ssL has time-dependent coefficients which present

computational difficulty when (2.1) is being used to solve for the phase quantities

directly. To obtain the phase currents from the flux linkages, the inverse of the

time-varying inductance matrix will have to be computed at every time step. The

computation of the inverse at every time step is time-consuming and could produce

numerical stability problems [10].

26

as

as′

rθ

cs

bs′ cs′

bs

q axis d axis

a axis

b axis

c axis

Fig. 2.2. Q-axis leading d-axis and the rotor angle represented as . rθ

To remove the time-varying quantities in voltages, currents, flux linkages and

phase inductances, stator quantities are transformed to a d-q rotating reference frame.

This results in the voltages, currents, flux linkages, part of the flux linkages equation and

inductance equations having time-invariant coefficients. In the idealized machine, the

rotor windings are already along the q- and d-axes, only the stator winding quantities

need transformation from three-phase quantities to the two-phase d-q rotor rotating

reference frame quantities. To do so, Park’s Transformation is used to transform the

stator quantities of an AC machine onto a d-q reference frame that is fixed to the rotor,

with the positive d-axis aligned with the magnetic axis of the rotor which has a

permanent magnet in PMSMs. The positive q-axis is defined as leading the positive

d-axis by in the original Park’s Transformation, as shown in Fig. 2.2. The original

Park’s Transformation equation is of the form:

/ 2π

27

[0 0 ( )dq dq r abc ]f T fθ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ (2.10)

where the d-q transformation matrix is defined as:

0

2 2cos cos( ) cos( )3 3

2 2( ) sin sin( ) sin( )3 3

1 1 12 2 2

r r r

dq r r r rT

π πθ θ θ

πθ θ θ θ

⎡ ⎤⎢ ⎥− +⎢ ⎥⎢ ⎥⎢⎡ ⎤ = − − − − +⎢⎢ ⎥⎣ ⎦ ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

23π ⎥

⎥ (2.11)

and its inverse is given by

1

0

cos sin 12 2( ) cos( ) sin( ) 13 3

2 2cos( ) sin( ) 13 3

r r

dq r r r

r r

T

θ θπ

θ θ θ

π πθ θ

−

⎡ ⎤⎢ ⎥−⎢ ⎥⎢ ⎥⎢⎡ ⎤ = − − −⎢⎢ ⎥⎣ ⎦ ⎢ ⎥⎢ ⎥⎢ ⎥+ − +⎢ ⎥⎣ ⎦

π ⎥⎥

⎤⎥⎦

(2.12)

In Fig. 2.2, if the d-q axis is turned radian degrees clock wise (CW), the q-d

transformation is obtained which is still considered a Park’s Transformation as shown in

Fig. 2.3. In the q-d transformation, the q-axis still leads the d-axis but angle

representation is changed such that now is the rotor angle between the q-axis and

a-axis which is chosen as a 0 reference for simplicity. Q-d transformation is used to

analyze the self and mutual inductances because of the ease of calculation. is

considered as the actual rotor position (generally it can be obtained by using an encoder

or similar position sensors) with respect to the a-axis which is used in the original Park’s

Transformation matrix, , but the angle depicted in Fig. 2.3, , which is

between the q-axis and a-axis in q-d transformation does not represent the real rotor

position. Therefore, the d-axis should be considered as a reference for measuring the real

rotor angle with respect to position selected as the a-axis in general. Eventually, the

d- and q- axis synchronous inductances will be independent of the rotor position so that

either of the transformations will give the same result.

/ 2π

θ

rθ

0 ( )dq rT θ⎡⎢⎣ θ

0

28

as

as′

θ

cs

bs′ cs′

bs

q axis

d axis

a axis

b axis

c axis

Fig. 2.3. Q-axis leading d-axis and the rotor angle represented as θ .

The Park’s Transformation which has a d-q reference is of the form:

[0 0 ( )dq dq r abc ]f T fθ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ (2.13)

where f represents either a voltage, a current or a flux linkage; 0 0, , t

qd q df f f f⎡ ⎤ ⎡=⎢ ⎥ ⎢⎣ ⎦ ⎣⎤⎥⎦

]

,

[ ] [ , , tabc a b cf f f f= .

The q-d transformation matrix is obtained as

0

2 2cos cos( ) cos( )3 3

2 2( ) sin sin( ) sin( )3 3

1 1 12 2 2

qdT

π πθ θ θ

πθ θ θ θ

⎡ ⎤⎢ ⎥− +⎢ ⎥⎢ ⎥⎢⎡ ⎤ = −⎢⎢ ⎥⎣ ⎦ ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

23π ⎥+ ⎥ (2.14)

29

And the inverse of the above transformation matrix is given by

1

0

cos sin 12 2( ) cos( ) sin( ) 13 3

2 2cos( ) sin( ) 13 3

qdT

θ θπ π

θ θ θ

π πθ θ

−

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢⎡ ⎤ = − −⎢⎢ ⎥⎣ ⎦ ⎢ ⎥⎢ ⎥⎢ ⎥+ +⎢ ⎥⎣ ⎦

⎥⎥ (2.15)

Basically, if the rotor position θ is known, the above matrix transforms the

variables of the stator reference frame to the one fixed to the rotor as shown in (2.13).

The relationship between θ and can be expressed with the help of Figs. 2.2

and 2.3 as

rθ

2rπ

θ θ= + (2.16)

Equation (2.16) depicts that the actual rotor position, , is 90 (electrical degree)

behind . If the q-d transformation matrix,

rθ

θ 0 ( )qdT θ⎡ ⎤⎢ ⎥⎣ ⎦ , is used as a transformation matrix,

the above description should be taken into account for deriving the q-d voltage equations

of a PMSM.

If is known, the above formula can be substituted into the q-d transformation

matrix with the help of some trigonometric reduction equations shown below:

rθ

0 ( )qdT θ⎡⎢⎣⎤⎥⎦

cos( ) sin2

sin( ) cos2

r r

r r

πθ θ

πθ θ

+ =−

+ = (2.17)

As a consequence of these manipulations, it has been observed that and

are basically the same, except for the ordering of the d and q variables. One

might say that is more useful if no additional modification is wanted for

assigning the actual rotor position, , as shown in Fig. 2.1.

0 ( )qdT θ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( )dq rT θ⎡⎢⎣⎤⎥⎦

⎤⎥⎦0 ( )dq rT θ⎡⎢⎣

rθ

30

Since ssL is analyzed according to the q-d rotating reference frame, the

corresponding relationship between flux linkages, 0qd⎡ ⎤Λ⎢ ⎥⎣ ⎦ , the q-d axis currents, 0qdi⎡ ⎤⎢ ⎥⎣ ⎦ ,

and the time-invariant q-d axis inductances, 0qdL⎡ ⎤⎢ ⎥⎣ ⎦ , can be obtained by transforming only

the stator quantities, that is:

[ ]1

0 0 0 0 0( ) ( ) ( )qd qd ss qd qd qd rT L T i Tθ θ θ−⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤Λ = + Λ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

(2.18)

where term is the transformation matrix which transforms the

phase inductance matrix,

1

0 0( ) ( )qd ss qdT L Tθ−⎡ ⎤ ⎡⎢ ⎥ ⎢⎣ ⎦ ⎣ θ ⎤⎥⎦

ssL , into the q-d axis synchronous inductance matrix, 0qdL⎡⎢⎣⎤⎥⎦

0

. If

the above description is applied to (2.18) then it can be shortened to:

0 0qd qd qdL i⎡ ⎤ ⎡ ⎤ ⎡Λ =⎢ ⎥ ⎢ ⎥ ⎢⎣ ⎦ ⎣ ⎦ ⎣⎤⎥⎦

⎤⎥⎦

(2.19)

by

0 0

0 0

0 0

, ,

, ,

t

qd q d

qd q d

t

qd q d

L L L L

i i i i

λ λ λ⎡ ⎤ ⎡Λ =⎢ ⎥ ⎢⎣ ⎦ ⎣⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

(2.20)

After simplifying the right hand side of (2.18), the rotor angle, , independent

q-d axis flux linkages, and , synchronous inductances,

rθ

qλ dλ qL and , and currents,

and are obtained as follows:

dL qi

di

0

0

0 0

3 ( )23 ( )2

( )

q ls ms q q q

d ls ms d r q q

ls

L L L i L i

L L L i L i

L i zero in balanced systems

λ

λ λ

λ

⎛ ⎞⎟⎜= + − =⎟⎜ ⎟⎜⎝ ⎠⎛ ⎞⎟⎜= + + + = +⎟⎜ ⎟⎜⎝ ⎠

=

rλ (2.21)

where

31

q ls m

d ls m

L L L

L L L

= +

= +q

d

(2.22)

In equation (2.22), mqL and are the common mutual inductances on the

d-axis and q-axis circuits, respectively. Traditionally, the sums, (

mdL

ls mqL L+ ) and

( ), are referred as the q-axis and d-axis synchronous inductances, respectively. ls mdL L+

Flux linkages in (2.20) can also be written as:

0

0

q

qd d

λλλ

⎡ ⎤⎢ ⎥

⎡ ⎤ ⎢ ⎥Λ =⎢ ⎥⎣ ⎦ ⎢ ⎥⎢ ⎥⎣ ⎦

(2.23)

On the contrary, flux linkages using the original Park’s Transformation is differ

from the above representation, 0 , , t

dq d qλ λ λ0⎡ ⎤ ⎡ ⎤Λ =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ . The only difference between

and is the order of and . When and are known either 0qd⎡Λ⎢⎣

⎤⎥⎦⎤⎥⎦0dq

⎡Λ⎢⎣ qλ dλ qλ dλ 0qd⎡ ⎤Λ⎢ ⎥⎣ ⎦

or can be constructed easily. Once their components are known the transfer

between each flux linkage matrix is easy.

0dq⎡Λ⎢⎣

⎤⎥⎦

]

So that (2.18) could have been written also as follows:

[1

0 0 0 0 0( ) ( ) ( )dq dq r ss dq r dq dq r rT L T i Tθ θ θ−⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤Λ = + Λ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ (2.24)

To collect the correct result out of the above equation, the rotor angle, θ , in the

inductance matrix, ssL , should be in the form of to be used in (2.24) by (2.16). After

computing the above formula, the stator d-q axis flux linkages, and , will be the

same as provided by (2.18). Results of the derivation of d-q axis stator flux linkages are

given in (2.21).

rθ

qλ dλ

The following part is the derivation of the d-q axis motor voltages of PMSM:

32

Equations (2.7) and (2.14) are the starting point. First, the original d-q Park’s

Transformation is applied to the stator quantities shown in (2.7). This is given

by:

0 ( )dq rT θ⎡⎢⎣⎤⎥⎦

[ ]

[ ]

[ ]

0 0

0 0

0 0

( )

( )

( )

dq dq r s

dq dq r s

dq dq r s

v T v

i T i

T

θ

θ

θ

⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎡ ⎤ ⎡ ⎤Λ = Λ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

(2.25)

where

0 , , t

dq d qv v v v⎡ ⎤ ⎡=⎢ ⎥ ⎢⎣ ⎦ ⎣ 0⎤⎥⎦

1

0

− ⎤Λ ⎥⎦

(2.26)

When the three-phase system is symmetrical and the voltages form a balanced

three-phase set of abc sequences, the sum of the set is zero, hence the third components

of the d-q quantities in (2.25) are zero, e.g. = 0. 0v

If (2.25) is substituted into (2.7), then the stator voltage equation is written in d-q

coordinates as:

[ ]1

0 0 0 0 0 0( ) ( ) ( ) ( )dq dq r s dq r dq dq r dq r dqv T r T i T p Tθ θ θ θ−⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ (2.27)

where, is the differential operation, p dpdt

= .

In a three-phase balanced system, all the phase resistances are equal,

= = s a br r r r= c

0⎤⎥⎦

1

0

− ⎤⎥⎦

, such that the resistive drop term in the above equation reduces to

[ ]1

0 0 0( ) ( )dq r s dq r dq s dqT r T i r iθ θ−⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡=⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ (2.28)

The second term having a derivative part in (2.27) can be expanded as:

( ) 1 1

0 0 0 0 0 0 0( ) ( ) ( ) ( ) ( )dq r dq r dq dq r dq r dq dq r dqT p T T p T T pθ θ θ θ θ− −⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡Λ = Λ + Λ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣

(2.29)

Further manipulations lead us to:

33

1 1

0 0 0 0 0 0 0 0( ) ( ) ( ) ( ) ( ) ( )dq r dq r dq dq r dq r dq dq r dq r dqT p T T p T T T pθ θ θ θ θ θ− −⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡Λ = Λ + Λ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣

1

0

− ⎤⎥⎦

0⎤⎥⎦

(2.30)

And finally,

( )1 1

0 0 0 0 0 0( ) ( ) ( ) ( )dq r dq r dq dq r dq r dq dqT p T T p T pθ θ θ θ− −⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡Λ = Λ + Λ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ (2.31)

where, the term ( )1

0 0( ) ( )dq r dq rT p Tθ θ−⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ is simplified as follows:

First, taking the derivative of (2.12), we get:

1

0

sin cos 02 2( ) sin( ) cos( ) 03 3

2 2sin( ) cos( ) 03 3

r r

dq r r r r

r r

p T

θ θπ

θ ω θ θ

π πθ θ

−

⎡ ⎤⎢ ⎥− −⎢ ⎥⎢ ⎥⎢⎡ ⎤ = − − − −⎢⎢ ⎥⎣ ⎦ ⎢ ⎥⎢ ⎥⎢ ⎥− + − +⎢ ⎥⎣ ⎦

π ⎥⎥ (2.32)

where, is the rotor angular speed in electrical radians/sec, rωr

rddtθ

ω = . Multiplying

equations (2.11) and (2.32), we obtain:

( )11 12

1

0 0 21 21

31 32

02( ) ( ) 03

0dq r dq r

T TT p T T T

T Tθ θ ω

−⎡ ⎤⎢ ⎥

⎡ ⎤ ⎡ ⎤ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎢ ⎥⎢ ⎥⎣ ⎦

(2.33)

where the elements of the matrix are written and manipulated by using some

trigonometric reduction equations. The results are given below [1]:

34

11

2 2 212

2 2 221

22

2 2 2 2cos sin cos( )sin( ) cos( )sin( ) 03 3 3 3

2 2 3cos cos ( ) cos ( )3 3 2

2 2 3sin sin ( ) sin ( )3 3 2

2 2 2 2sin cos sin( ) cos( ) sin( ) cos(3 3 3

r r r r r r

r r r

r r r

r r r r r r

T

T

T

T

π π π πθ θ θ θ θ θ

π πθ θ θ

π πθ θ θ

π π πθ θ θ θ θ θ

=− − − − − + + =

=− − − − + =−

= + − + + =

= + − − + + +

31

32

) 03

1 2sin sin( 2 / 3) sin( ) 02 31 2 2cos cos( ) cos( ) 02 3 3

r r r

r r r

T

T

π

πθ θ π θ

π πθ θ θ

=

⎧ ⎫⎪ ⎪⎪ ⎪=− + − + + =⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭⎧ ⎫⎪ ⎪⎪ ⎪=− + − + + =⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭

(2.34)

Putting all the elements obtained in (2.34) into (2.33), we get:

( )1

0 0

30 02

2 3( ) ( ) 0 03 2

0 0 0

dq r dq rT p Tθ θ ω−

⎡ ⎤⎢ ⎥−⎢ ⎥⎢ ⎥⎢ ⎥⎡ ⎤ ⎡ ⎤ = ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(2.35)

More simplification leads us to:

( )1

0 0

0 1 0( ) ( ) 1 0 0

0 0 0dq r dq rT p Tθ θ ω

−⎡ ⎤−⎢ ⎥

⎡ ⎤ ⎡ ⎤ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎢ ⎥⎢ ⎥⎣ ⎦

(2.36)

Equations (2.31) and (2.36) yield the following result, written in expanded matrix

form:

0 0 0

0 1 01 0 00 0 0

d d d

q s q r q

v iv r i pv i

λ λω λ

λ λ

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤−⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥= + +⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦0

d

qλ (2.37)

The above matrix can be arranged in equation form shown below:

35

( )

d s d r q d

s d r q q d d

s d d r q

v r i p

r i L i L pi

r L p i L i

ω λ λ

ω

ω

= − +

= − +

= + − q

(2.38)

Similarly,

( ) ( )

q s q r d q

s q r d d r r q q

s q q r d d r

v r i p

r i L i L pi

r L p i L i

ω λ λ

ω ω λ

ω λ

= + +

= + + +

= + + +

(2.39)

Equations (2.38) and (2.39) can be put together to form the famous d-q voltage

model for the PMSM in matrix form. This is given in below:

01

d s d q r dr r

q d r s q q

v r L p L iv L r L p i

ωλ ω

ω

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤+ −⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥ + ⎢ ⎥+ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦

(2.40)

To obtain the d-q reference frame electromechanical torque, , and the

instantaneous input power are used. Details are as follows:

emT

inP

Total input power into the machine can be represented as inP

in sa sa sb sb sc scP v i v i v i= + + (2.41)

When the stator phase quantities are transformed to the rotor d-q reference frame that

rotates at a speed of , (2.41) becomes: /r rd dω θ= t

(32in q q d dP v i v i= + ) (2.42)

where the zero sequence quantities are neglected. Using (2.42), the mechanical output

power can be obtained by replacing and with the associated speed voltages as outP dv qv

(32out r q d r d qP iω λ ω λ= − + )i (2.43)

36

For a P-pole machine, , where is the mechanical rotor speed in

radians/sec, produced electromechanical torque is derived when the output power

is divided by the mechanical speed , which is given as:

( / 2)r Pω = rmω rmω

emT

outP rmω

( )

( )

32 23 ( )2 2

em d q q d

d q q d r q

PT i i

P L L i i i

λ λ

λ

= −

= − + (2.44)

It is apparent from the above equation that the produced torque is composed of

two distinct mechanisms. The first term corresponds to “reluctance torque” due to the

saliency (difference in the d-axis and q-axis reluctance or inductance), while the second

term corresponds to the “excitation torque” occurring between and the permanent

magnet [10].

qi

rλ

2.2. Introduction to Brushless DC (BLDC) Motors

Brushless DC (BLDC) motors contain most of the characteristics of PMAC

motors, explained in Section 2.1.1. They are generally designed as surface-mount motors

with concentrated windings on the stator. With the appropriate placement of the

permanent magnet on the rotor, a trapezoidal back-EMF shape is obtained.

As compared to a conventional DC brush motor, brushless DC (BLDC) motors

are DC brush motors turned inside out, so that the field is on the rotor and the armature

is on the stator. As explained before, field excitation of BLDC motor is provided by a

permanent magnet and commutation is achieved electronically instead of using

mechanical commutators and brushes.

One of the main characteristics of the BLDC motor is that it is a modified PMSM

such that the back-EMF is trapezoidal instead of being sinusoidal. To achieve smooth

torque production, quasi-square wave currents are applied to the stator phases when the

37

corresponding back-EMFs have a flat constant portion (approximately 120 degrees like

the currents). The “commutation region” of the back-EMF of a BLDC motor should be

as small as possible. On the other hand, when driven by a Current Source Inverter that

region should not be extremely narrow making it difficult to commutate a phase of the

motor.

Each phase of a BLDC motor is energized by a digital controller in a sequence.

The energization is synchronized with the rotor position in order to produce constant

torque, therefore, it is important to know the rotor position in order to understand which

winding will be energized following the energizing sequence. One alternative way to

sense the position of the rotor is by using an optical position sensor. Optical position

sensors consist of phototransistors (sensitive to light), revolving shutters and a light

source. The output of an optical position sensor is usually a logic signal. This is

especially useful when unipolar switching is used to drive a BLDC motor [5].

The most common method of sensing in a BLDC motor is using hall-effect

position sensors. For a brushless DC (BLDC) motor with a trapezoidal back-EMF, it is

sufficient to get position information that is updated at each 60 degree electrical interval.

The position information is used then by the microcontroller/DSP to decide the

triggering of inverter switches. Generally, three hall-effect sensors are used for a three

phase motor, (they will be named as Hall-A, Hall-B, and Hall-C each with 120 degree

lag with respect to the previous sensor), to determine the position of the rotor magnetic

field.

Hall-effect sensors are generally embedded into the stator on the non-driving end

of the motor. According to the corresponding “Hall Effect Theory”, named after E.H.

Hall who discovered it in 1879, whenever the rotor magnetic poles pass near the

hall-effect sensors, a high or low signal will be generated indicating when the N (north)

or S (south) pole is passing near the sensors. In general, the North Pole signal

corresponds to “1” and the South Pole to “0” [11]. In motors not having a neutral

accessibility each hall-effect sensor output signal is synchronized with the

corresponding line-to-line back-EMF for alignment easiness. For example in Fig. 2.4

38

Hall-A is aligned with and so forth and so on. This means that the output

of each hall-effect sensor is actually leading the zero crossing of the corresponding phase

of each back-EMF by 30 electrical degrees.

ab a bE E E= −

ci

bi

ai

aE

bE

cE

aH

bH

cH

abE

aH aH

-30 0 30 60 60 120 150 180 210 270240 360330300 906030 180150120 240210

tω

tω

tω

tω

tω

tω

tω

Fig. 2.4. Line-to-line back-EMF ( ), phase back-EMFs, phase current waveforms and

hall-effect position sensor signals for a BLDC motor [3].

abE

39

For example, as illustrated in the digital output of Fig. 2.4, is leading by

30 degrees. In motors that have neutral access, each of the hall-effect sensor signals can

represent the corresponding phase back-EMF. Alignment can then be easily achieved

since there is a neutral connection available to observe the phase back-EMF waveforms

to be used for the alignment process.

aH aE

The motor that is used in this thesis, a 48 pole 36 slot PMSM, has that kind of

luxury such that its hall-effect signals are aligned with the corresponding phase

back-EMF voltages. Fig. 2.4 shows an example of the hall-effect sensor signals with

respect to the corresponding line-to-line back-EMF voltages and phase currents. For

example, to know the starting point of the phase A current, the digital output of

alone is not sufficient; additionally the falling edge signal of is needed to trigger the

switch T1 (T6 is still on) [5].

aH

cH

As a consequence, when the hall-effect signals generate a logic of ”1 0 0”,

“ ”, respectively, T1 is turned on while T6 still conducts to energize the phase

A current shown in Fig. 2.4. By observation, the logic high of a hall-effect position

sensor is maintained for 180 electrical degrees. This can be a significant disadvantage,

since timers may have to be used in order to reduce the switching times to 120 electrical

degrees (two-thirds of the hall-effect position sensor output high of 180 electrical

degrees) [5].

aH bH cH

2.2.1. Mathematical Model of BLDC Motor

In general, the analysis of a BLDC motor is conducted in phase variables due to

its non-sinusoidal back-EMF and current. Fig. 2.5 shows the equivalent circuit of a

BLDC motor and a power inverter. The analysis of the BLDC motor drive is based on

the following assumptions for simplification [6]:

40

• The motor is not saturated,

• The motor windings have a constant resistance, self inductance, and

mutual inductance. The resistance and inductance of all phases are

identical,

• All three phases have an identical back-EMF shape,

• Power semiconductor devices in the inverter are ideal,

• Iron losses are negligible,

• Eddy current and hysteresis effects are neglected.

+-

+ -

+-

ar cr

br

aaLccL

bbL

aece

be

M

M MDC-Link Capacitor

a

b

c

T1 T3 T5

T4 T6 T2

Fig. 2.5. Equivalent circuit of the BLDC drive (R, L and back-EMF) [3].

The model of a BLDC motor consisting of three phases, as shown in Fig. 2.5, is

explained by the following series of equations. Since there is no neutral used, the system

is wye connected, thus the sum of the three phase currents must add up to zero, i.e.

0sa sb sc

sa sb sc

i i ii i i

+ + =

+ =− (2.45)

Under the above assumptions and referring to Fig. 2.5, a three-phase BLDC

motor mathematical model can be represented by the following equation in matrix form:

41

0 00 00 0

sa a sa aa ba ca sa a

sb b sb ab bb cb sb b

sc c sc ac bc cc sc c

v r i L L L i edv r i L L L idt

v r i L L L iee

⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥= + ⎢⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎥ +⎥⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦

(2.46)

If the rotor has a surface-mounted design, which is generally the case for today’s

BLDC motors, there is no saliency such that the stator self inductances are independent

of the rotor position, hence:

aa bb ccL L L= = = L (2.47)

Again with no saliency all mutual inductances will have the same form such that,

ab ba bc cb ca acL L L L L L M= = = = = = (2.48)

Also, under balanced three-phase condition all the phase resistances are equal, such that:

s a br r r r= = = c (2.49)

Equations (2.47-2.49) are substituted into (2.46), giving the following reduced matrix

form,

0 00 00 0

sa s sa sa a

sb s sb sb b

sc s sc sc c

v r i L M M i edv r i M L M idt

v r i M M L i

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

ee

+ (2.50)

Using the concept of (2.47), the inductance matrix above is simplified and the

resultant matrix form is given by:

0 0 0 00 0 0 00 0 0 0

sa s sa sa a

sb s sb sb b

sc s sc sc c

v r i L M i edv r i L M idt

v r i L M i

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤−⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥= + −⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

ee

+ (2.51)

where the back-EMF waveforms , and are trapezoidal in nature and can be

represented by either Fourier Series or Laplace Transforms.

ae be ce

After rearranging the equations, the model of the BLDC motor depicted in

Fig. 2.5 is reduced to Fig. 2.6 which resembles (2.51).

42

+-

+ -

+-

ar cr

br

L M−ae

cebeDC-Link

Capacitor

a

b

c

T1 T3 T5

T4 T6 T2

L M−

L M−

Fig. 2.6. Simplified equivalent circuit of the BLDC drive (R, (L-M) and back-EMF).

The electromechanical torque is expressed as

a a b b c cem

m

e i e i e iT + +=

ω:

where

ae is the phase-to-neutral back-EMF of phase A in volts,

be is the phase-to-neutral back-EMF of phase B in volts,

ce is the phase-to-neutral back-EMF of phase C in volts,

sai is the stator current in phase A in ampere,

sbi is the stator current in phase B in ampere,

sci is the stator current in phase C in ampere, and

rmω is the rotor angular velocity in mechanical radian/sec; , ( / 2)re rmPω ω=

P is the number of poles,

reω is the rotor angular velocity in electrical radian/sec.

The Laplace domain representation of the BLDC motor model will lead us to

construct the block diagram of the overall BLDC motor system easily. Since the

back-EMF waveforms are periodic, they can be represented by a periodic function [5].

43

The voltage equation of a BLDC motor, which is the same as brush DC motor

model, can be depicted in the time domain by substituting (2.51) in (2.52) shown below:

( )( ) ( ) ( ) aa a e rm a

di tv t r i t k t Ldt

ω= + + (2.52)

where

( )v t is the supply voltage (line-to-neutral) in volts,

ai is the armature current in ampere,

aL is the armature inductance; , aL L M= −

( )e rmk ω t is the line-to-neutral back-EMF, shown as E ,

ek is the back-EMF constant in Vs/rad.

The Laplace form of (2.52) is given as:

( )( ) e rma

a a

V s kI sr sL

− Ω=

+ (2.53)

By using the following equations, the transient model of a BLDC motor can be

obtained as illustrated in Fig. 2.5.

If the current term is put aside in (2.52), we obtain

( ) ( ) ( )( )a aa

a a a

di t rv t E ti tdt L L L

= − − (2.54)

where is the periodic back-EMF which is a function of the rotor angle. ( ) e rmE t k ω=

The relation between the electrical power, , and the mechanical power, , is: eP mP

e a m emP Ei P T rmω= = = (2.55)

The electromechanical torque, , is linearly proportional to the armature current , i.e. emT ai

em T aT k= i (2.56)

where is the torque constant. Equation (2.55) can then be substituted into (2.56)

yielding:

Tk

44

em arm

ETω

= i , (2.57)

such that for the ideal motor with no saturation, resistance, or voltage-drops in the

controller:

e rmt

rm r

kEk ekωω ω

= = ≈ (2.58)

Provided that and are in consistent units (such as and ), it

can be seen from the derivations above that the ideal squarewave brushless DC (BLDC)

motor has the same basic energy-conversion equation as the DC commutator motor.

Tk ek /Nm A /Vs rad

The motion equation is:

( ) ( ) rmem L rm

dT t T t J Bdtω

ω= + + (2.59)

where, , , and LT J B stand for load torque, inertia and friction, respectively.

The motion equation can be expressed in Laplace form as:

( )em L

rmT TB sJ

−Ω =

+ (2.60)

Equation (2.56) can also be expressed in Laplace form as:

( ( ) )T eem

a a

k V s kTr sL

− Ω=

+rm (2.61)

45

2.3. Mechanical Structure of Permanent Magnet AC Motors

There are mainly two types of PMAC motor construction which are based on the

way magnets are mounted on the rotor and the shape of the back-EMF waveform that

was explained briefly above. In case where the magnets are mounted on the rotor

surface, the PMAC motor is called surface-mounted. If the magnets are buried inside the

rotor then it is called an interior-mounted motor. The back-EMF waveform can be either

sinusoidal or trapezoidal as expressed above. Detailed information about the PMAC

motor construction is as follows:

1) Surface-mounted PMAC Motor:

The typical design of this type of PMAC motor is shown in Fig. 2.7. Its name

implies that the magnets are placed on the surface of the rotor. Doing so makes the

mounting of the magnets on the rotor easier and less costly. Moreover, skewed poles can

be easily magnetized on the round rotor to minimize cogging torque. Since the magnets

are located on the rotor surface, the machine has a large effective air-gap, which makes

the effects of saliency negligible (thus the direct-axis magnetizing inductance is equal to

the quadrature-axis magnetizing inductance, md mq mL L L= + ).

Furthermore, because of the large air-gap, synchronous inductance

( s sl mL L L= + ) is small and therefore the effects of armature reaction are negligible. A

further consequence of the large air-gap is that the electrical time constant of the stator

winding is small. One problem with this type of design is that during high-speed

operation magnets can detach from the rotor [4].

There could be various shapes of magnets for surface-mounted PMAC motor.

Bar-shaped and circumferential segments with angles up to 90 degrees and thickness of a

few millimeters are available. Radially magnetized circumferential segments produce a

smoother air-gap flux density and less torque ripples [4].

46

N

N

N

N

S

S

S

S

d-axis

q-axis

Fig. 2.7. Cross sectional view of four pole surface-mounted PM rotor ( d qL L= ).

2) Interior-mounted PMAC Motor:

The typical construction of this type of PMAC motor is shown in Fig. 2.8. Each

permanent magnet is mounted inside the rotor core rather than being bonded on the rotor

surface; the motor not only provides mechanically robust and rugged construction, but

also has possibility of increasing its torque capability. This type of PMAC motor can be

used for high-speed applications since the magnets are physically contained and

protected. By designing a rotor magnetic circuit such that the synchronous inductances

vary as a function of rotor angle, the reluctance torque can be produced in addition to the

excitation torque (which is the multiplication of q-axis current and the rotor flux linkage

due to permanent magnet) [4].

47

The reluctance torque is generated by this kind of motor because each magnet is

buried and covered by a steel pole piece. This significantly changes the magnetic

behavior of the machine since, via these iron pole pieces, high-permeance paths are

created which are between two consecutive magnet poles as shown in Fig. 2.8. This path

is considered as the quadrature axis in which its inductance is higher than the inductance

measured through the direct axis (also shown in Fig 2.8). One can say that the q-axis is

leading the d-axis by 90 electrical degrees, thus there are saliency effects which

significantly alter the torque production mechanism of the machine [4].

As described in Section 2.1.1, in addition to the excitation there is a reluctance

torque which is due to the difference in the d- and q-axes inductances, the so called

saliency. This class of interior PM (IPM) synchronous motors can be considered as a

combination of the reluctance synchronous motor and the surfaced-mounted PM

synchronous motor. Currently, it is very popular in industrial and military applications

for providing high power density and high efficiency compared to other types of electric

motors [6].

N

S

N

S

N S NS d-axis

q-axis

Fig. 2.8. Cross sectional view of four pole interior-mounted PM rotor ( d qL L≠ ).

48

Another classification for PMAC motors is based on the shape of the back-EMF

generated (either as trapezoidal or sinusoidal). These back-EMF waveforms can be

obtained by a generator test in which open-circuit voltages are induced in the stator

windings due to the permanent magnets which show the characteristics of the back-EMF

[6].

If the flux distribution and back-EMF shapes are trapezoidal then this type of

motor is called a brushless DC (BLDC) motor, whereas if the shape of back-EMF

waveform is sinusoidal then this is considered a PM synchronous motor (PMSM). More

details about these types of PMAC motors are discussed in below:

3) PMAC Motors with Sinusoidal Shape Back-EMF:

For the PMAC motor with a sinusoidal back-EMF, the so called PMSM, all three

stator phases are excited at one instant of time providing quasi-sinusoidal current flow

through the stator windings. Since all of the stator phases are excited simultaneously, the

torque per ampere ration is higher than PMAC motors with a trapezoidal shape

back-EMF.

In this type of PMAC motor, stator conductors are sinusoidally distributed for each

phase to achieve a sinusoidal current waveform. Also, permanent magnets are arranged

such that they generate a sinusoidal flux distribution in the air-gap. With the interaction

of sinusoidal current and sinusoidal flux, and because of the lack of a commutation

scheme like BLDC motors, ripple free torque is generated.

The commutation in PMSM motors is not going to be as harsh as the BLDC

motor which is fed by square wave currents. The condition in which a change of cycle of

square wave currents occurs, huge torque ripples are generated. On the other hand, for

the sinewave condition there is a smooth transition from one cycle to another.

The construction of this type of PMAC motor is more expensive than BLDC

motors. Additionally, PMSM drives use continuous motor position feedback to feed the

motor with sinusoidal voltage or current. Due to this reason, expensive position sensors

49

(>100$) like the resolver or encoder are necessary, which increases the overall cost of

the motor drive system.

Unlike the BLDC trapezoidal shape back-EMF motors, the working principle of

a PMAC motor with a sinusoidal back-EMF is very similar to the AC synchronous

motor, therefore it can be analyzed with a phasor diagram.

4) PMAC Motors with Trapezoidal Shape Back-EMF:

The stator currents of a BLDC motor are quasi-square wave with 120

electrical-degree conduction periods. Magnetic flux distribution in the air-gap is

rectangular. When the phase currents are timed approximately with the constant part of

the back-EMF a constant torque is generated. These rectangular current-fed motors have

concentrated windings on the stator. Unlike the PMSM which needs continuous position

sensing, PMAC motors with trapezoidal shape back-EMFs (BLDC), require sensing

only at every 60 electrical-degree period for commutation of the phase currents. As a

result, six commutation points are required for one electrical cycle [6].

Basically, in each commutation point, the switching cycle is changed. By doing

so, there is no need to have a high resolution position sensor such as an absolute or

incremental encoder or resolver. In this case, a very simple and cheap (<1$) hall-effect

sensor would be appropriate enough. During these commutation intervals, however

torque ripples occur. Minimizing the torque ripples is an on-going research topic that is

not in the scope of this thesis.

2.4. Advantages and Disadvantages of PMAC Motors

The following are some of the most common advantages of PMAC motors over

other electric motors available on the market:

• PMAC motors are high efficient motors. PMSM and BLDC motors, are

considered the most efficient of all electric motors. There is an absence of

50

a rotor circuit, instead of which permanent magnets on the rotor are used

for the excitation (generating constant rotor magnetic field). These

magnets consume almost no power whatsoever, so copper losses are

negligible at the rotor level unlike AC induction and synchronous motors.

Also, the friction is lower and the durability is higher, since there are no

mechanical commutators and brushes to wear out. All of these

characteristics of PMAC motors put them into the top spot of the high

efficiency category [6].

• Recent advances in high-energy density magnets, such as samarium

cobalt and Neodymium Iron Boron (NdFeB), have allowed achievement

of very high flux densities in the PMAC motors. These magnets provide

providing high torques and allow for the motors to be built smaller and

lighter [6].

• One category of PMAC motors, BLDC motors, can be controlled with a

closed-loop system as simple that is as a DC brush motor because the

control variables are easily accessible and constant throughout the

operation of the motor. PMSM, on the other hand, has an open-loop

control capability unlike BLDC motors in which its closed-loop control is

not as easy as the BLDC motor, but is similar to the induction motors.

PMSM drives need advanced control techniques, such as vector control

and direct torque control, just like AC induction motors [6].

• In PMAC motors, there is no rotor circuit available, reducing losses. The

only heat is produced on the stator, which is easier to cool than the rotor

because the stator is static and is located on the outer part of the motor

[6].

• PMAC motors have low maintenance, great longevity and reliability.

Without brushes and mechanical commutators, regular maintenance is

greatly reduced and risks such as sparking in explosive or corrosive

environments like oil field etc are eliminated. The longevity of the motor

51

is mainly based on the winding insulation and bearings (like other electric

motors), and the life-length of the magnet [6].

• There is no noise generated with the commutation because it is achieved

electronically, not mechanically. The converter switching frequency is

high enough so that the noise created by the harmonics is not audible [6].

PMAC motors also have some inherent disadvantages just like any other

electrical machine. These are summarized in the following:

• Magnet cost is the most important issue in PMAC motors. Rare-earth

magnets such as samarium-cobalt and neodymium boron iron are

especially costly. Neodymium boron iron is the most powerful, but is the

most expensive. If the initial cost is a major issue in applications where

PMSM motors would be used, then the cost of higher energy density

magnets prohibits their use in those kinds of applications [6].

• Very large opposing magneto motive forces (MMF) and high temperature

can demagnetized the magnets. Although the critical demagnetization

force is different for each magnet material, extra caution must be taken to

cool the motor, especially if it is built compact [6].

• For surface-mounted PMAC motors, high speed operation is limited or

not possible because the mechanical construction of the rotor. The rotor is

not suitable to handle huge centrifugal forces at high speed. The magnets

are adhered onto the rotor. This type of assembly is not strong and

compact enough not to hold the magnets under high speed circumstances.

On the other hand, interior-mounted PMAC motors have the capability to

run at high speeds with no problem on the rotor side because the magnets

are buried inside the rotor, but demagnetization still is an important factor

in those types of motors [6].

• There is a limitation in the range of the constant power region, especially

for surface-mounted PMAC motors. In the constant power region, the so

called “flux weakening” region, the flux of the rotor is weakened to

52

extend the speed torque curve as far as possible. Surface-mounted PMAC

motors have no saliency so that flux generation current does not appear in

the torque equation to weaken it. Interior-mounted PMAC motors have

the saliency effect causing the d-axis (direct axis) current to be weakened

which extends the speed versus torque curve and eventually the constant

power region (flux weakening region). PMAC motors with trapezoidal

shape back-EMFs (BLDC motor) are incapable of achieving a maximum

speed greater than twice the base speed [6].

• Because there is a constant energy on the rotor due to the permanent

magnets, PMAC motors present a major risk in case of short circuit

failures in the inverter. If any short circuit happens in the inverter during

the operation of the motor, the rotor constantly induces an electromotive

force in the short-circuited windings causing a very large current to flow

through those windings. This produces large torque which tends to block

the rotor. For vehicle applications, the danger of blocking one or several

wheels of a vehicle is not acceptable [6].

53

2.5. Torque Production Comparison of PMSM and BLDC Motors

The torque productions are for each motor compared in the following paragraphs,

assuming the same copper loss.

The BLDC motor has a high torque density in low and medium speeds but is not

suitable for extended high-speed operation because of its limited flux weakening

capability. Also, in BLDC operation with a rectangular current wave shape, torque

pulsation occurs at the 6th harmonic of the fundamental frequency.

The flowing equations show the torque density difference between BLDC motors

and PMSMs in the constant torque region (the PMSM has superiority in the

flux-weakening region over the BLDC motor):

Assume psI and pI are the peak values of the stator currents in the PMSM and

BLDC machines, respectively. The rms values of these currents are:

2ps

sy

II = (2.62)

23

pd

II = (2.63)

Equating the copper losses and substituting for the currents in terms of their peak

currents gives:

23 3 2sy s d sI R I R= (2.64)

Therefore,

32p psI I= (2.65)

The ratio of output torque is obtained from these relationships as follows:

(2 ) /_ 1.1547_

3 /2 2

p p

p ps

xE xIBLDC TorqueE IPMSM Torque

x x

ω

ω=⎛ ⎞⎜ ⎟⎝ ⎠

= (2.66)

54

The above result shows that the BLDC motor has approximately a 15.5% greater

torque compared to the PMSM in the constant torque region [12]. A PMAC with a

sinusoidal back-EMF (PMSM) can be operated as a BLDC. In ideal commutation, the

current is conducting 120 electrical degrees with its center aligned with the peak point of

the back-EMF, as shown in Fig. 2.9. On the other hand, a brushless DC motor can also

be operated as a PM synchronous motor as depicted in Fig. 2.10, but this type of control

creates more torque ripples. The motor used in this thesis, a 48 pole 36 slot exterior rotor

pancake type PMSM, generates a sinusoidal back-EMF.

In this thesis, the proposed method discussed regulates the motor speed using a

control scheme similar to typical BLDC motor control scheme without current regulation

with the help of three cost-effective hall-effect sensors. Since the motor is

surface-mounted and generates sinusoidal back-EMF, it is considered as surface-

mounted PMSM. Vector control techniques can be used so that a relatively ripple-free

average torque is obtained.

As a consequence, if the torque ripple is not a big deal, a simple trapezoidal

current control scheme (BLDC type), instead of a complicated sinusoidal current control

scheme (PMSM type), can be a simple, cost-effective and quite efficient solution for

washing machine speed control applications.

55

Back-EMF

Phase current

ωt

Fig. 2.9. Ideal BLDC operation.

Back-EMF

Phase current

ωt

Fig. 2.10. PMSM controlled as BLDC.

2.6. Control Principles of the BLDC Motor

+-

+ -

+-

ar cr

br

L M−

aece

beDC-Link Capacitor

a

b

c

T1 T3 T5

T4 T6 T2

L M−

L M−

BLDC Motor Model

PositionD2D4

D3D1dL

sV

Fig. 2.11. Voltage Source Inverter (VSI) PMAC drive [13].

56

Almost all home appliances operate on a single-phase input supply. Fig. 2.11

illustrates the circuit configuration of a widely used PMAC motor drive in low-cost

applications, such as washing machine control. This configuration is commonly used

today in both BLDC and PM synchronous motor (PMSM) drives. The only differences

between the drives are the control methods used and the choice of position sensing if no

sensorless technique is used. The BLDC motor drive which is represented in Fig. 2.12

consists of an uncontrolled diode rectifier converting single phase AC to DC, a DC-link

electromagnetic capacitor for energy storage, and a Voltage Source Inverter (VSI). The

DC-link energy is distributed using the part of VSI, made up of IGBTs or MOSFETs,

based on PWM scheme. That is why this type of drive is also called a “PWM inverter

PMAC motor drive.” [13]

There are various types of PWM techniques which can be used in AC motor

drives. The main goal is to reduce the harmonic content to a minimum and achieve the

best utilization of the DC bus. Also, an expensive EMI filter is inevitable at the input to

comply with various utility interface regulations. When a sensorless technique is not

used, one of the necessary components of a BLDC drive is a position sensor. Either a

hall-effect sensor or an optical shutter arrangement is used along with some sort of

microcontroller/microprocessor or Digital Signal Processor (DSP) to control the VSI.

Although snubbers are not shown, in practical applications they are necessary to protect

the switches from voltage spikes generated by switching [14].

+-

+ -

+-

ar cr

br

aaLae

cebeDC-Link

Capacitor

a

b

c

T1 T3 T5

T4 T6 T2

ccL

bbL

PMSM Model

PositionD2D4

D3D1dL

sV

Fig. 2.12. Voltage Source Inverter (VSI) BLDC drive.

57

A profile for each phase of the BLDC motor with respect to the corresponding

back-EMFs as a function of the rotor position is shown in Fig 2.4. At each rotor position,

the constant current multiplies with the constant part of the back-EMF to give torque;

hence the sum of the products of a phase back-EMF and the corresponding phase current

is constant such that this result produces constant torque which is represented in (2.52).

The desired current profile is achieved by supplying the BLDC motor from either a VSI

or a Current Source Inverter (CSI) which is not discussed here. When using a VSI, the

desired current profile is achieved by controlling the switching of the transistors [14]. At

any given time, only two of the six switches are conducting. This means that only two

phases are conducting at any instant and the third one is floating. In other words, as

current enters one of the three phases and leaves through the other and there is no current

flowing in the third phase. Having no current in the third phase helps in sensing the

back-EMF for a sensorless control of BLDC motor.

The convention of negative and positive current schemes represented in this

thesis is as follows the current entering any phase of a motor is assigned as a positive

sign, and the current that is leaving any phase of a motor is assigned as a negative sign.

Therefore, the upper switches of the inverter, namely, T1, T3 and T5, carry a current

(flowing into the motor) which is assigned as a positive sign. The lower switches of the

inverter, namely, T2, T4 and T6, carry a negative current (flowing out from the motor).

In order to properly operate this motor, it is necessary to synchronize the phase currents

with the corresponding phase back-EMFs. This is achieved by the use of hall-effect

position sensors which detect the position of the rotor magnetic field, and hence the

position of the rotor shaft. In conclusion, a motor with synchronized switching produces

an average positive torque.

As seen in Fig. 2.4, each phase current conducts and turns off in a definite

sequence. For both the positive and negative cycles on the same phase there is 120

electrical degrees of conduction and 60 electrical degrees of turn off. The current

commutation from one phase to the other corresponding to a particular state of the

back-EMF is synchronized by the hall-effect sensors such that for every 60 electrical

58

degrees of rotation one of the hall-effect sensors changes state. Given this, it takes six

steps to complete an electrical cycle. Basically, for every 60 electrical degrees, the phase

current switching should update.

One electrical cycle, however, may not correspond to a complete mechanical

revolution of the rotor. The number of electrical cycles to be repeated to complete a

mechanical rotation is determined by the rotor pole pairs. For each rotor pole pairs, one

electrical cycle must be completed. The number of electrical cycles/rotations is then

equal to the number of rotor pole pairs.

A typical closed-loop block diagram of a brushless DC (BLDC) motor drive

system with phase current control is shown in Fig. 2.13. Similar to DC brush motors,

BLDC motors with electronic commutation can operate from a DC voltage source and

the speed can be adjusted by varying supply voltages. Higher voltage means higher

speed. This can be achieved by rectification of a variable voltage transformer output or

by using a thyristor bridge in which the DC bus voltage is a function of the firing angle

of the thyristor bridge. Instead of varying supply voltage, variable speed operation can

also be realized from a constant bus voltage. Fig. 2.11 represents a typical configuration

of this phenomenon in which power semiconductor switches are used not only to

commutate but also to control the motor terminal voltage via the PWM (Pulse Width

Modulation) technique.

59

DC-Link Capacitor

a

b

c

T1 T3 T5

T4 T6 T2

BLDCMotor

Commutation Logic

PI + +-

SpeedRegulator

CurrentRegulator

VSI

PositionSensor

ReferenceSpeed

CurrentSensors

ω*ω

Fig. 2.13. A typical current-controlled BLDC motor drive with position feedback [7].

PWM generates a fixed frequency (usually 2 kHz – 30 kHz) voltage pulse whose

on-time duration can be controlled. As a rule of thumb, the PWM frequency should be

at least 10 times that of the maximum frequency of the motor. When the duty cycle of

the PWM signal is varied within sequences, the average voltage supplied to the stator

changes, thus the motor speed will change. If the duty cycle is higher, the motor will turn

faster; if it is smaller the motor will slower. Another advantage of having PWM is that if

the DC bus voltage is much higher than the motor rated voltage, the motor can be

controlled by limiting the percentage of the PWM duty cycle corresponding to that of the

motors rated voltage. This adds flexibility to the controller to hook up motors with

different rated voltages can be connected to the drive and be utilized at the rated voltage

by controlling the PWM duty cycle [14].

60

Since the brushless DC motor is highly inductive, the motor current produced

from the switched voltage will be almost identical to that of a fixed voltage source

whose magnitude is the average of the switched voltage waveform. Although PWM

control is now very popular in drives, variable bus voltage control is still used in some

applications where dynamic performance is not important [14].

A B C

dI

dI

A B C

dI

dI

A B C A B C

dI

dI

A B C A B C

dI

dI

dI

dI

dI

dI

Mode 1

Mode 3 Mode 6

Mode 2 Mode 5

Mode 4

Fig. 2.14. Six possible switching sequences for a three-phase BLDC motor [15].

There are six possible switching modes in the inverter for a three-phase BLDC

motor, illustrated in Fig. 2.14. The power circuit of the electronic controller is a

switchmode circuit. The only means of controlling such a circuit is to control the timing

of the gate signals that turn the power transistors on and off.

61

According to the quasi-squarewave current condition (120 electrical degrees) in

BLDC motors, only two inverter switches (one in the upper inverter bank and one in the

lower) are conducting at any instant, as shown in Fig.2.14. If the motor is connected in

wye, this means that the inverter input current, dI , is only flowing through two of the

three motor phases in series at all times. Only a single current sensor in the DC-link is

sufficient to accomplish the current regulation of the current, dI , which flows through

two phases. With the help of three hall-effect sensors, the position of all the three phase

currents can be known so that the proper switches are turned on/off using the PWM

current control mode through hysteresis current controllers [15].

For instance, when the currents (equal in magnitude) tend to exceed the

hysteresis band, both switches are turned off at the same instant to initiate current

feedback through the diodes. Finally, each phase current is regulated and synchronized

with the corresponding back-EMF to achieve desired torque and speed characteristics.

S1 S3

S4S2

D1

D2

D3

D4

dVsC + -

abEabL

D0

I

II

S1, S4 ON S1, S4 ONS1, D3

ONS1, D3

ON

FrewheelingEnergy delivery from

sourceON

OFF

HI

LI

0t Ht LtHT LT

IΔ

Time (s)Phas

e C

urre

nt

Switc

h St

ate

I II

Fig. 2.15. PWM current regulation for BLDC motor [15].

62

Current regulation for motoring action, using the inverter switches corresponding

to mode 1 (switches S1 and S6 are turned on) from Fig. 2.14, is shown in Fig. 2.15.

When switches S1 and S6 are turned on, current dI flows through phase A and B. For

this system, line-to-line inductance 2( )abL L M= −

E E E= −

and line-to-line back-EMF

are used (displayed in Fig. 2.15) instead of phase to neutral values.

Resistance drops are neglected.

ab a b

When the applied source is bigger than the back-EMF voltage, , current

flows through the motor when switches S1 and S6 are turned on; otherwise if is less

than , then current might flow from the motor. If one of the switches in Fig. 2.15 is

turned off (let’s say S6) then current decays due to the back-EMF and it flows through

diode D3 and switch S1 (loop II), short-circuiting the motor terminals. This is called

freewheeling. The amplitude of the phase current is regulated by controlling the duty

cycle of the switch S6 (chopping) such that

dV abE

dV

abE

IΔ is kept at a constant minimum value.

The simple BLDC drive is a current-regulated drive in which there is only one

current reference signal that is used for all six pairings of the transistors. As it can be

seen in Fig. 2.13, the current reference signal is derived from the speed error signal,

which represents the difference between the desired speed and the actual speed. The

control system for the BLDC motor shown in Fig. 2.13 has three hall-effect sensors for

position sensing along with three hysteresis current regulators. Its simple version is

explained above, for each phase, and the PI speed regulators are combined together to

form the fundamental speed control scheme of a three-phase BLDC motor [16].

2.7. Control Principle of the PM Synchronous Motor

In the three-phase PMSM control, more sophisticated control such as vector or

direct torque control techniques are required because all the three phases are conducting

at the same time unlike BLDC motor control. A sinewave drive generates less than 1%

63

torque ripple as compared to a squarewave drive (BLDC drive), but this is only true

when both the back-EMF and phase current have sinusoidal waveforms. Because current

in each phase is a sinusoidal function of rotor position, separate PWM control for each

individual phase currents is required.

E

si

sVrλ

rθ

qi

di

q-axis

d-axis

Phase A-axis

Fig. 2.16. Basic phasor diagram for PMSM [15].

Vector control, or field-oriented control, for all sinewave drives is fundamentally

based on the two-axis theory and the phasor diagram shown in Fig. 2.16. Vector control

is appropriate for sinewave motors in which torque is produced by the interaction of the

fundamental flux and the fundamental ampere-conductor distribution. There are several

different ways of creating a two-axis system but generally the method operates with the

d- and q- axis current phasor components, and , which may be defined in a variety

of reference frames such as rotor or fixed to the stator. To determine and i from the

instantaneous line currents, a reference frame transformation, such as Park’s

Transformation shown in Fig. 2.11, is used [15].

di i

i

q

d q

Referring to the instantaneous phasor diagram in Fig. 2.16, responsive current

regulation plus self-synchronization make it possible to place the instantaneous current

phasor si anywhere within dq-axis rotor reference frame under the limitation of

maximum current and regulator saturation constraints. Independent control of the

64

orthogonal current components and is the simplest known way to control the output

torque for PMSMs, shown in (2.44) [15].

di qi

PositionSensor

PMSM

ai∗

bi∗

ci∗

di∗

qi∗

df

qf

( )rT θ

rθiφ

Current-regulatedVSI

Vectorrotatord-q current

program

CommandTorque

emT ∗

Fig. 2.17. Basic block diagram for high-performance torque control for PMSM [15].

The baseline for implementing this type of high-performance torque control for

sinusoidal PMAC motors (PMSM) is shown in Fig. 2.17. An incoming torque command,

, is extracted into the d- and q-axis currents by using (2.44). These are shown as the emT ∗

df and qf function blocks in Fig. 2.17. These current commands, and , in the

dq-axis rotor reference frame are DC quantities for a constant torque command. To

apply sinewave currents to the motor, these quantities are then transformed into the

instantaneous sinusoidal current commands (

di qi

aI , bI , and cI ) for the individual stator

phases using the rotor angle feedback ( rθ ) and the inverse reference frame

transformation matrix as in (2.12). 1−⎤⎥⎦0 ( )dq rT θ⎡⎢⎣

Commonly, the df and qf functions map the torque command into the dq-axis

current component commands, and , so that the maximum torque per ampere

operation is achieved providing maximized operating efficiency. There are two different

current phasor,

di qi

si , trajectories that lie in the synchronously rotating dq-axis:

1) Maximum torque-per-amp trajectory for non-salient (surface-mounted) PMSM

which is represented in Fig. 2.18.

65

2) Maximum torque-per-amp trajectory for salient (interior-mounted) PMSM (so

called IPMSM) which is plotted in Fig. 2.19.

As it can be seen in Fig. 2.18, torque-per-amp in a non-salient PMAC machine is

maximized by setting to zero regardless of the torque value, so that the sinusoidal

phase currents are always exactly in phase (or 180 degrees out of phase) with the

sinusoidal back-EMF voltages.

di

( )di A

rλ

emT+

increasing

emT+

decreasing

increasingdecreasing

incr

easi

ngde

crea

sing

0

0

Salient poletrajectory

si


( )di A

rλ

emT+increasing

emT+decreasing

increasingdecreasing

incr

easi

ngde

crea

sing

0

0

siSamplevector

Salient poletrajectory

si

45°


66

This phenomenon resembles the BLDC motor control that is described in

Section 2.11. The idea of creating a positive torque is by aligning the phase currents with

the corresponding back-EMF such as in surface-mounted PMSM control. As a

consequence, there is a similarity between surface-mounted PMSM control and BLDC

motor control. In contrast, the maximum torque-per-amp is achieved for salient PMAC

machines (interior-mounted PMSM) by changing both and by using the di iq df and qf

mapping functions obtained with respect to torque values [15].

67

CHAPTER III

MEASUREMENT OF THE PM SPINDLE MOTOR PARAMETERS

3.1. Introduction

All instantaneous torque-controlled drives (vector drives, direct-torque-controlled

drives) use various machine parameters. Motor parameters must be considered carefully

in order to facilitate a proper simulation platform to resemble the real system. The main

objective of this chapter is to show how to measure the necessary motor parameters used

in the simulations. There are many techniques which can be used for the identification of

the PMSM parameters without direct measurement; the most useful one is called

self-commissioning with identification which will be discussed briefly in Section 3.5,

but this technique is not used in this thesis.

Temperature rise is a very difficult parameter to calculate unless the thermal time

constant has been measured. Under heavy load conditions temperature rises and all the

parameters that are measured change proportional to the temperature, however, in this

thesis temperature change effect will not be considered on the motor parameters in the

simulations.

Because the motor used in the thesis is a high torque, low speed, 48 pole, outer

rotor and inner stator PMSM and the operational speed of the agitation control is

between 0 and 100 rpm (which is less than the base speed (~150 rpm) of the motor). It is

assumed that the motor will not go through the field-weakening region where the

saturation effects on the dq-axis synchronous inductances must be taken into account. To

model the real motor in the simulations the following parameters should be measured:

• Resistance ( sR ) and phase inductance ( saL : Phase A self-inductance)

• Back-EMF constant ( ) ek

68

3.2. Principles of Self-Commissioning with Identification

In many industrial applications, the AC machine and converter are sold by

separate manufacturers and the parameters of the AC machine are not known. Prior to

starting the variable-speed drive system, however, some machine parameters have to be

known. Although the name-plate of the machine can be used for this purpose, in many

applications this does not yield accurate information on the parameters required. In

general, extensive testing of the machine and tuning of the controller is required, but this

can be expensive and time-consuming and requires specially trained equipment (which

also increases the costs).

It is possible to obtain various machine parameters and to tune the controllers by

an automated process during a self-commissioning stage. It is important to note that

various modern industrial drives now incorporate self-commissioning techniques. For

example, in a permanent magnet synchronous machine the required electrical parameters

such as stator resistance and dq-axis synchronous inductances can be obtained from

on-line measurements of the stator currents and/or stator voltages when the machine is at

standstill while the inverter in the drive is utilized to generate the signals required for the

parameter estimation. [16]

In self-commissioning with identification, various test signals (step signals, ramp

signals, etc.) are applied to the machine and the responses are measured. It is also

possible to use the name-plate of the motor and other data to ensure that the machine

will not be destroyed during the tests. It is also a goal to use a minimum number of

sensors. Currents are always measured in every drive to ensure safe operation of

semiconductor switches, even when a sensorless technique is being used. However, it is

not desirable to use too many sensors. Additional voltage and/or speed sensors should be

avoided in self-commissioning systems [16].

There are basically two types of Self-commissioning with identification.

1) Off-line identification: Test signals are applied to the motor and the

response data is measured. This is used to identify the parameters.

However, it is a disadvantage that parameter variations are not considered,

69

since these can only appear during the normal operation of the drive. When

this approach is used it is also a goal not to produce torque [16].

2) On-line identification: In this case, initial pre-computed controller

parameter values are required. Measurements are made during the

operation of the drive and motor and controller parameters are

continuously estimated, and the controller is continuously adapted. This

approach imposes high computation burdens [16].

3.3. Resistance and dq-axis Synchronous Inductance Measurements

The phase resistance of the motor is measured by a HP 4284A Precision LCR

Meter (20 Hz ~ 1 MHz) at two different frequencies (20 Hz and 400 Hz). This is because

when the frequency is increased resistance also increases due to the skin effect. Other

than the RLC meter, precision ohmmeters or multimeters can also be used for the

measurement of the phase resistance. For our purpose, an RCL meter will give more

accurate results than the rest of the methods because the test frequency can be adjusted

relative to our needs. Because the operation speed range of the tested motor is between 0

and 100 rpm (0 Hz to 40 Hz), the test frequency is set to 20 Hz in the RLC meter which

is equal to the average value of the speed range of agitating cycle. On the other hand, a

test conducted with the frequency 400 Hz in the phase resistance measurement is

conducted because it could be used in future high speed PMSM washing machine

applications. At high speed (high frequency) the resistance will change due to skin

effect.

In this thesis work, a dq-axis model (SIMPLORER® has a built-in model of

PMSM) of the motor in simulation platforms (MATLAB/SIMULINK® and

SIMPLORER®) is built because the tested motor generates a sinusoidal back-EMF. This

ensures that the dq-axis model is applicable even if the motor is supplied by squarewave

currents like a BLDC motor. Knowing the phase resistance, sR , therefore is important in

70

the dq-axis motor model. The wye connected motor has a neutral access, so the

measurement of the resistance is done between the phase and neutral point as shown in

Fig. 3.1. If the three-phase wye connected motor does not have a neutral access then the

only way of measuring the phase resistance is by measuring the line-to-line resistance,

LLR , and dividing it by 2. Consequently, the line-to-line resistance, LLR , is two times the

phase resistance, sR , when the motor is three-phase symmetrical.

Usually in three-phase motors, the windings are connected in wye fashion and

there is no neutral point to access. Therefore for those types of motors the inductance

measurement is conducted between two phases (line-to-line inductance) which is given

by

2 2(LL sa sb abL L L M L M= + − = − ) (3.1)

where saL = sbL = L is the phase self-inductance, (Phase A and Phase B self inductances

are equal under the three-phase symmetrical system) and abM is the mutual inductance

between Phase A and Phase B.

In the above equation, the line-to-line inductance of Phase A and B is

represented. Under symmetrical systems, ( saL = sbL = scL = L and abM = acM = bcM = M )

the above equation can be applicable to the other remaining phases and the resultant

line-to-line inductance for the corresponding two phases will be always be equal

to 2( )L M− .

In our case, the motor has a neutral access such that each phase inductance can

be measured between phase and neutral point by the HP 4284A Precision LCR Meter

(20 Hz ~ 1 MHz) at both 20 Hz and 400 Hz test frequencies. The test level is chosen as

1 V which is generally chosen as a common test level for motor inductance

measurements.

In sinewave PMSM drives generating sinusoidal back-EMFs, the parameters of

interest are the per-phase resistance, sR , and the per-phase synchronous inductance, sL ,

(both of which are employed in constructing the phasor diagram which is the main tool

71

for analyzing sinewave operation are necessary). Accordingly, the appropriate methods

for measuring synchronous inductance should be used. The synchronous inductance

incorporates the self and mutual inductances in a way that correctly represents their

effects with a sinewave drive [16].

As mentioned in the previous chapter, phase inductance is represented in (2.5). In

this case, the q-axis in dq-plane is chosen as a reference for the rotor position with

respect to the a-axis as shown in Fig. 2.3. This is done because deriving the phase

inductance is easier if the q-axis is taken as a reference for the rotor position, θ . In

general motor drives, the d-axis is the reference to represent the rotor position with

respect to the a-axis as shown in Fig. 2.2. If the rotor position, θ , is rewritten

incorporating the real rotor position, , in (2.5), then new phase inductance when the

d-axis is chosen as a reference point for the rotor position, , is obtained as:

θ

rθ

rθ

0 cos 2sa sl msL L L L θ= + + r (H) (3.2)

The above equation demonstrates the self-inductance of Phase A. The only

difference between (3.2) and (2.5) is that the sign of is now positive because the

dq-axis is turned 90 electrical degrees CCW, and now represents the angle between

the d-axis and the a-axis (the a-axis is the 0 degree reference).

msL

rθ

72

δ

d-axis

q-axis

s sI R

E

s sjI X

V

sIϕ

rλ

Fig. 3.1. Vector phasor diagram for non-salient PMSM.

The synchronous inductance can be measured by several methods such as a

short-circuit generating test for non-salient machines and a lock-rotor test by using the

RLC meter as shown in Fig. 3.2. In this thesis the later one is used for simplicity because

the first method requies an additional motor that needs to be coupled so that generation

is possible. In the short-circuit generating test the motor is driven at a low speed with all

the phases shorted together. The current in one phase, or preferably in all three, is

measured. If a non-salient motor is wye connected in steady-state with balanced

sinusoidal phase currents, the operation can be represented by the phasor diagram shown

in Fig. 3.1, and then its voltage equation can be written as:

( s s )R j Lω= + +V E I (3.3)

where V , E and are represented in bold showing that they are vectors and rotate in

the dq-plane. is the synchronous speed in rad/sec, V is the applied voltage phasor,

I

ω E

is the back-EMF phasor and is the current phasor. In a sinewave permanent magnet I

73

synchronous machine, the d-axis and q-axis synchronous inductances are equal;

ds qs sL L= = L . The voltage, back-EMF and the current phasors are described below:

The terminal voltage phasor is given by

ds qsV jV= +V (3.4)

And the current phasor is:

ds qsI jI= +I (3.5)

The d-axis is chosen as the real, or reference axis. The open-circuit voltage

phasor, (the back-EMF phasor) is aligned with the q-axis in the phasor diagram since it

leads the flux-linkage by 90 electrical degrees. It can be written as:

qs rjE jωλ= =E (3.6)

where is the RMS phase flux-linkage due to the magnet. rλ

The current dI flows in the short-circuited motor which is equal to (3.7) when

and if the resistance is neglected in (3.3), then: 0V =

sSC

ELI

ω = (3.7)

where E is the value of the open-circuit voltage per phase at the same speed. Caution

should be taken during the short-circuit test because the short-circuit current may be

much larger than the rated current, and may be sufficient enough to partially

demagnetize the magnet. Also, short-circuit connection should not be applied suddenly

while the machine is running; first it should be connected while the machine is stationary

and then the machine should be brought to the desired test value slowly. The reason is

that a sudden short-circuit produces a higher transient current with DC-offsets that might

demagnetize the magnets [17].

The first method explained above is only applicable in the non-salient motors.

For salient motors, the short-circuit generation test will help to measure only the d-axis

synchronous inductance. Other tests should be conducted to get the q-axis inductance.

74

Together it is a time-consuming and complicated method. The second technique used in

this thesis to measure the synchronous inductance is an easy solution as compared to the

first method which is explained below:

According to Fig. 3.2, self-resistance, sR , and self-inductance, saL , are measured

by the LCR meter for two different frequency levels (20 Hz and 400 Hz) as discussed

before. Because the self-inductance and mutual inductance are dependent of the rotor

position, the rotor dependent terms in (3.2) should be eliminated. To achieve this, the

rotor magnetic flux (d-axis) is aligned (locked) with the a-axis. The alignment is

obtained by applying a DC power supply between Phase A and the neutral access of the

motor which is also illustrated in Fig. 3.2. The above condition should be considered

when the motor has saliency, meaning that the d- and q- axis inductances are not equal

such that the self and mutual inductances are dependent on the rotor position. In our

case, the motor (non-salient = surface-mounted PMSM) has very little saliency so that

the saliency effect can be neglected. Actually, from the above conclusion, to measure the

d- and q-axis synchronous inductances for surface-mounted PMSMs there is no need to

align the magnet flux with the known reference axis (a-axis). The saliency effect

coefficient, , in the self and mutual inductance equations will be zero for non-salient

motors.

msL

asR

bsR csR

asL

bsL csL

Neutral

A-Phase

B-Phase C-Phase

After aligning the rotor flux with Phase A using DC power supply, LCR meter was used to

measure the self-inductance and self-resistance of Phase A.

Fig. 3.2. Measurement of phase resistance and inductance.

75

When the rotor magnetic flux (d-axis) is aligned with the a-axis then in (3.2)

becomes zero and the self-inductance of Phase A can be written as

rθ

0sa slL L L L= + + ms (3.8)

where is the dq-axis synchronous inductance variation. Since msL2

dms

L LL

−= q in

surface-mounted motors and slL are equal to zero, then in steady-state saL can be shorten

to:

0saL L L= + ms (3.9)

When = 0 then (3.9) can be reduced farther to: msL

0saL L= (3.10)

From the simplified self-inductance above a relationship is found between the

d- and q-axis synchronous inductances. It is obtained as follows:

ds md ls

qs mq ls

L L LL L L

= +

= + (3.11)

where slL is chosen as zero under steady-state conditions and the d- and q-axis

synchronous inductances are equal to their corresponding magnetizing values and mdL

mqL , respectively as shown below:

03 ( )2md ms dsL L L L= + = (3.12)

03 ( )2mq ms qsL L L L= − = (3.13)

If = 0 is placed in (3.12) and (3.13), we obtain msL

032md mq ds qsL L L L= = = = L (3.14)

76

If (3.10) is substituted into (3.14), then the d-axis synchronous inductance

relationship with the a-axis self-inductance can be gathered as:

03 32 2ds qs saL L L= = = L (3.15)

Self-resistance and self-inductance measurements of the 36 slot/48 pole spindle

motor at 20 Hz and 400 Hz are given in Table I below:

TABLE I MEASUREMENTS OF SELF-RESISTANCE AND SELF-INDUCTANCE

Ω (ohm) @ 20 Hz (test level 1V)

mH @ 20 Hz (test level 1V)

asR : 16.285 Ω

bsR : 16.31 Ω

csR : 16.3345 Ω

20savg HzR : 16.30983 Ω

aaL : 62.09 mH

bbL : 62.04 mH

ccL : 61.10 mH

20savg HzL : 61.7433 mH

Ω (ohm) @ 400 Hz (test level 1V)

mH @ 400 Hz (test level 1V)

asR : 21.11 Ω

bsR : 21.2115 Ω

csR : 21.10 Ω

400savg HzR : 21.1405 Ω

aaL : 61.72 mH

bbL : 62.10 mH

ccL : 61.86 mH

400savg HzL : 61.8933 mH

As a result of the measurements of the three-phase self-resistances and

self-inductances of the tested motor, the following conclusions are obtained about the

average self-resistance and self-inductance of the motor to be used in simulation

platforms (MATLAB/SIMULINK® and SIMPLORER®):

_avg selfR : 16.30983 Ω (chosen from 20 Hz test)

_avg selfL : 61.8183 mH (chosen from average of 20 Hz and 400 Hz tests)

77

The synchronous inductance of the motor is obtained using (3.13) and the above

average value of the self-inductance from above as:

3 3 61.8183 92.72745 2 2s ds qs saL L L L= = = = = mH

Although using the LCR meter is an easy way to measure the line-to-line and

line-to-neutral (if the motor has a neutral access) inductances, LCR meters generally use

a test frequency on the order of 1 kHz with very low currents, so the measured

inductance may differ appreciably from the correct value which corresponds to a lower

frequency and much higher currents. A complete experimental analysis of the

inductances requires DC inductance measurements using, for example, the Prescot

bridge described by Jones [17], which can be modified to permit inductances to be

measured with varying levels of bias to demagnetizing stator MMF or quadrature-axis

stator MMF [17].

It is a good idea to take inductance measurements at several rotor positions.

Unless there are some problems with the magnets in the motor, there should be very

little inductance variation observed as the rotor rotates in a surface-mounted PMSM (our

test motor is a surface-mounted PMSM). On the other hand, interior-mounted PMSMs

have a saliency such that inductances vary greatly with the rotor position. When the

d-axis inductance is measured by a locked-rotor test in an interior-mounted PMSM the

q-axis synchronous inductance can be measured if the rotor is rotated 90 mechanical

degrees CCW or CW from the d-axis. The rotor could have been brought to the 90

degree electrical position when the a-axis is chosen as a reference, but if the motor is a

high pole motor such as our test motor, rotating the motor 90 electrical degrees from the

reference position is more difficult than rotating it 90 mechanical degrees.

To get the synchronous inductance of the tested motor more precisely another

method can be performed which does not eliminate the leakage inductance in the

calculations. This technique is applicable if the motor has a neutral access. In this thesis,

this method is not implemented, but the theory behind it will be explained in below:

78

To find the leakage inductance, slL , a line-to-line inductance measurement

should be performed corresponding equation is given by:

2LL sa sb abL L L M= + − (3.16)

where abM is the mutual inductance between Phase A and Phase B. If the system is

symmetrical then all the self-inductances are equal,( ), therefore the

line-to-line inductance can be summarized as:

sa sb scL L L= = = L

2( )LL abL L M= − (3.17)

If the line-to-line inductance is measured between Phase A and Phase B and the

self-inductance measurement result, which is obtained from the previous experiment for

Phase A is placed into (3.17), then the mutual inductance, abM , between Phase A and

Phase B can be found easily.

01 cos 22ab ms rM L L θ=− + (3.18)

Because the dq-axes inductances are the same, the saliency coefficient for mutual

inductances will also be equal to zero and the final mutual inductance equation, without

the term, can be shown as: msL

012abM L=− (3.19)

The self-inductance of Phase A without the term and including leakage

inductance is written as:

msL

0sa slL L L= + (3.20)

Knowing abM and putting it in (3.19), is obtained. If is then substituted in

(3.20), then

0L 0L

slL can be found because saL was known previously from the LCR meter

result which includes the leakage inductance effect also.

79

If the d- and q-axis inductances, sdL and sqL , in (3.12) and (3.13) are re-written

including the known leakage inductance, then we obtain:

03 ( )2ds md ls ms lsL L L L L L= + = + + (3.21)

03 ( )2ds mq ls ms lsL L L L L L= + = − + (3.22)

where all the components in the above equations are known so that the d- and q-axis

inductances, sdL and sqL , can be precisely found.

3.4. Measurement of the Back-EMF Constant ( ) and PM Flux Linkage ( ) ek rλ

To calculate the rotor flux linkage, rλ , which is the line-to-neutral peak rotor flux

linkage amplitude value, first the back-EMF constant, ek , is measured. This done by the

generator test in which the tested motor is coupled to a DC motor and it is driven

through the shaft by the DC motor at a constant mechanical speed, rmω . The flux linkage

of the permanent magnet, rλ , and the peak value of back-EMF constant, ek , can then be

easily obtained by measuring the no load line-to-neutral peak voltage (back-EMF, anpkE )

of the motor while it is driven by DC motor. The derivation of rλ is given by:

2 2 2anpk e re e eE k kω= = fπ (3.23)

If the peak back-EMF constant, 2 ek , is replaced by the line-to-neutral peak

rotor flux linkage value, , in (3.23), we obtain: rλ

2 0.223256 /anpkr e

re

Ek Vλ

ω= = = rad s (3.24)

80

where 2re rmP

ω ω= ; is the number of poles and is the mechanical speed in

rad/sec.

P rmω

When the motor has no neutral access, the measurement of the peak

line-to-neutral back-EMF, anpkE , is impossible. In this case, the following procedures

need to be conducted to find the peak rotor flux amplitude, : rλ

First, the line-to-line and line-to-neutral voltage relation is written as:

3ab anV = V (3.25)

The relationship between the rms and peak voltages can be expressed as:

2pk rmsV V= (3.26)

The measured line-to-line rms value of the back-EMF is equal to:

abrms e reE k ω= (3.27)

where is the line-to-line back-EMF constant. From (3.25), the back-EMF is

considered as a voltage, therefore:

ek

3ab

anEE = (3.28)

The peak value of (3.27) is given by:

2abpk e reE k ω= (3.29)

If (3.29) is substituted into (3.28), we obtain:

23

eanpk re

kE ω= (3.30)

Equation (3.30) is the placed into (3.24) and the following result is gathered.

2 2 23 3

anpk abpkr

re rm

E Ek

Pλ

ω ω= = = e (3.31)

81

TABLE II MEASUREMENTS OF LINE-TO-NEUTRAL BACK-EMF FOR EACH PHASE (0-50 RPM)

Ean (V-rms) Ebn (V-rms) Ecn (V-rms) freq-Ea(Hz) speed(RPM)0 0 0 0 0

4.417 4.467 4.68 4.56 11.34.397 4.513 4.648 4.52 11.34.321 4.33 4.524 4.38 11.34.384 4.514 4.626 4.53 11.38.2 8.159 8.348 8.3 20.8

8.256 8.207 8.429 8.31 20.88.24 8.216 8.41 8.34 20.87.993 8.58 8.396 8.35 20.811.701 11.887 12.106 12.03 30.411.914 12.038 12.211 12.18 30.411.996 12.119 12.243 12.25 30.412.04 12.17 12.31 12.19 30.416.34 16.28 15.75 16.15 40.216.28 16.25 15.71 16.18 40.216.25 16.25 15.7 16.14 40.216.08 16.11 15.58 15.96 40.215.56 16.3 16.3 16.15 40.215.53 16.27 16.31 16.11 40.219.67 19.79 19.65 19.85 5019.91 19.88 19.9 20.05 5019.91 19.88 19.9 20.04 5019.92 19.9 19.94 20.08 50

If the average value of the line-to-neutral back-EMF, , found in Table II,

is substituted into (3.23), then the average back-EMF constant, , can be derived as:

anpkavgE

ek

ek =0.223256 Vrad/s

3.5. Load Angle (δ ) Measurement

Load angle measurement is performed in this thesis to validate the motor model

in the simulations by making the motor model as similar to the real motor as possible.

82

δ , is the angle between the back-EMF,Load angle, E , and the input voltage of the

same phase back-EMF, V , as shown in Fig. 3.1.

To measure the load angle, the following equipment is used:

1) Three-phase AC power source

2) LeCroy Digital Storage Oscilloscope (DSO)

3) Maxtrol Hysteresis Dynamometer (max torque is 11.3 lbs)

4) Hall-effect sensor mounted on the stator

Back-EMF (Phase-A)

Hall-Effect Signal (Phase-A)

0δ =ωt

Fig. 3.3. Phase A voltage waveform and hall-effect signal of the same phase.

A load angle measurement is conducted for Phase A of the tested motor. It could

have been performed for the other phases, but under symmetrical conditions the results

would be the same. For the motor that it is used in this thesis, the line-to-neutral

back-EMF is aligned with the corresponding hall-effect signal so that hall-effect sensor

signals are used to get the back-EMF, E , position. The load angle will be the angle of

back-EMF, E , with respect to the input voltage of the same phase back-EMF, V . A

difficulty that exists though is reading the hall-effect signals on the oscilloscope

efficiently. This is due to jittering in the hall-effect signals. The reason this jittering

occurs is because of uneven distribution of the magnets on the rotor. Whenever the N

pole is seen by the hall-effect sensor it generates a pulse, and when the S pole is seen the

signal goes to zero. Fig. 3.4 shows an example of uneven distribution of the magnets

causing jittering in the hall-effect signals:

83

N SN S N S

15 V

0V

24 magnets in one electrical revolution.

Hall-effect signals

Time(s)

Fig. 3.4. Rotor magnet representation along with hall-effect signals.

From the above figure, it can be observed that uneven distribution of the magnets

alters the width of the hall-effect signals and causes jittering in the running motor as

mimicked in the graph below:

Back-EMF (Phase-A)

Hall-Effect Signal (Phase-A)

δ is not exactly zero because of jittering.

? t

Fig. 3.5. Phase A voltage waveform and hall-effect signal with jittering effect.

As it can be seen in Fig. 3.5, uneven distribution of the magnets on the rotor

causes shaking back and forth in hall-effect signals creating difficulty in reading those

signals on the oscilloscope. To overcome the jittering problem (not %100), The LeCroy

DSO is used to average 20 pulses of the hall-effect signal for Phase A, . aH

There are two different ways of measuring the load angle which are explained

below:

84

1) The motor is started and operated in open-loop by a three-phase AC power

supply. When V / f control is used starting from V / f = 1 (20 Vrms) up to 2

(40 Vrms) for a reference test frequency at 20 Hz, when operating frequency is

chosen as 40 Hz (100 rpm) using V / f ~= 1 (40 Vrms) created an unbalance

operation of the motor such that over 30 Hz the motor oscillates a lot (very

unbalanced current is observed after 25-30 Hz with V / f ~= 1). To diminish the

oscillation, the motor should have had a higher current flow through them over

the 25-30 Hz frequencies. In this case V / f ~= 2 is chosen which means there is

an 80 Vrms input sine wave applied to handle up to 10 N·m of load torque at

100 rpm (40 Hz). V / f ~= 2 is chosen as our operating point for most of the

tests to achieve less oscillation in the current, speed and torque response. The

results for the 40 Hz test frequency under no-load but with an increased applied

voltage condition is shown in Fig. 3.7.

The three-phase AC power supply is a programmable AC source such that

various voltage and frequency levels as well as steps in either can be preset. For

example, in the load test the motor is started at a low frequency, such as 12 Hz to

15 Hz, and then the frequency is automatically increased to the desired test level

(20 Hz and 40 Hz).

The applied voltage, V , is then increased from 20 Vrms up to 40 Vrms for the

first part of the load test (the second part starts from 70 Vrms and up to

90 Vrms). The voltage is incremented by 5 Vrms at the test frequency. At each

increment the average of the hall-effect signal is read on the DSO. The DSO then

calculates the load angle between the applied voltage and its corresponding hall-

effect signal. When the applied voltage is increased, the load angle increases

simultaneously, shown in the vector phasor diagram of the PMSM in Fig. 3.9.

When the applied voltage, sV , is increased, the current, sI , also increases so that

the sV vector moves farther than corresponding back-EMF, thus the load angle

85

increases. Related results to the first measurement technique are shown in

Figs. 3.6 and 3.7.

2) The above procedures are applied to the second part of the load test in which

motor is started under open-loop conditions as discussed before up to a 40 Hz

operating frequency with V / f =~ 2 (80 Vrms). When the motor reaches 40 Hz,

a load torque is applied from zero up to 10 N·m by the dynamometer. The results

are obtained as a graph shown in Fig. 3.8. When the motor is loaded, the current

flows through the terminal conductors to the motor and the applied voltage

deviates from the corresponding back-EMF so that the load angle is generated.

With the phase (radian) function and reasonable sweeping (25-100 sweeps), the

load angle, δ , values are collected for various operation points and are shown in the

figure below:

Experimental vs. Simulation Load Angle (deg) @ 20Hz Under No-Load(Ld=Lq=92.7274mH)

0

5

10

15

20

25

30

35

20 25 30 35 40 45

Applied voltage (RMS)

Delta

(deg

)

delta.exp delta.sim

Fig. 3.6. Experimental vs. simulated load angle (deg.) at 20 Hz (No-load).

86

Experimental vs. Simulation Load Angle (deg) @ 40Hz Under No-Load (

0.

5.

10.

15.

20.

25.

70 75 8 85 90

Delta

(eg

)

Ld=Lq=92.7274mH)

25

25

25

25

25

25

0 95

Applied voltage (RMS)

ddelta.exp delta.sim

Fig. 3.7. Experimental vs. simulated load angle (deg.) at 40 Hz (No-load).

Experimental vs. Simulation Load Angle (deg) @ 40Hz Under Load (Ld=Lq=92.7274mH)

-10

-5

0

5

10

delta.sim delta.exp Linear (delta.exp)

15

20

0 1 2 3 4 5 6 7 8 9 10 11

Load Torque (Nm)

Del

ta (d

eg)

Fig. 3.8. Experimental vs. simulated load angle (deg.) at 40 Hz (Load).

The reason we compared the load angles of both the simulation and experiment is

that for the same rms value of the motor current ( ), the behavior of the overall system

is different. This is because of the possible difference between the parameters of the

sI

87

motor model and the real motor. The overall behavior of the PMSM system is shown in

the graph below:

δ

d-axis

q-axis

s sI R

E

s sjI X

V

sIϕ

δ′

V

sI

Fig. 3.9. Possible vector representation of PMSM.

It can be shown from Fig. 3.9 that even if the amplitude of sI is same, the

difference in position of sI will affect the behavior of PMSM. That is why load angle

values from both the simulation and the experiment are compared.

88

CHAPTER IV

OPEN-LOOP SIMULINK® MODEL OF THE PM SPINDLE MOTOR

4.1. Principles of PMSM Open-Loop Control

When the stator resistance, sR , is neglected (near the motor base speed ), the

torque expression for non-salient (surface-mounted PMSM) is given in (5.7), referred to

as (4.1):

bω

3 s2em

e s

PTX

δω

⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜⎝ ⎠sV E

in Equation Section 4(4.1)

where is the variable angular frequency, 2eω efπ , of the input AC supply to the motor,

sV is the rms AC supply voltage of the motor at frequency ef and sX is the synchronous

reactance, 2 e sf Lπ , of the motor at the supply frequency.

Since the back-EMF, E , and the synchronous reactance, sX , are proportional to

the angular frequency , then the same maximum torque (when ) is achieved if

the

eω 90δ =

s

e

Vω

or s

e

Vf

ratio is maintained constant at all speeds (frequencies) at a given load

angle . This control method is very similar to the one used for induction motors. The

above methodology used as an open-loop control for PMSM drives is valid when the

motor speed is near the base speed (or base frequency). This is because near base speed

the back-EMF of the motor,

δ

E , and supply voltage, sV , are much larger than the voltage

drop in the stator resistance. On the other hand, at low frequencies the voltage drop due

to the stator resistance may not be negligible.

89

In open-loop control ( /V f control) the speed of the motor is commanded by the

input frequency ef of the supply, and sV is chosen such that at all speeds /V f is

maintained constant so as not to strengthen the stator flux linkage more than its normal

value. To implement the idea of /V f control, a sinusoidal PWM technique can be used.

Sudden reference speed changes may cause load angle to exceed its limit (=90 ) as a

precaution, a filter in the speed reference input may be used, as shown in Fig. 4.1.

Generally,

δ

sV is increased proportionally with ef until base frequency is reached.

Afterwards, the constant torque region ends and field weakening region, or constant

power region, is started. In that region sV is maintained constant at the base voltage, ,

of the motor providing constant power. Above the base speed, , only the supply

frequency

bV

bω

ef can be increased which weakens the stator flux linkage so that maximum

torque is reduced according to (5.7).

PMSM

A

B

C

ef

bV

bfsV

11 fT s+

t

Speed filterto avoid suddenspeed change

Speedreference

referencesV

referenceef

SPWMVSI

Fig. 4.1. Basic open-loop block diagram of PMSM.

90

4.2. Building Open-Loop Steady and Transient-State Models

Even though the conventional washing machine agitation speed control system is

simulated in the SIMPLORER® platform, open-loop steady state and even transient

controls are done in MATLAB/SIMULINK® because steady-state motor models are not

provided in SIMPLORER®. The ones it provides are transient models, and modification

of the models is not allowed. Therefore, relying on the convenience of SIMULINK®,

open-loop, steady-state and transient models are built in the MATLAB/SIMULINK®

platform. The dq-rotor model equations used in the open-loop control of the PMSM are

written again:

( )

( )q s s q r s d r

d s s d r s q

v R L p i L i

v R L p i L i

ω ω

ω

= + + +

= + −rλ

rλ

(4.2)

where is the derivative. p

If the derivative terms are removed, the steady-state voltage equations in the

rotor reference frame become:

q s q r s d r

d s d r s q

v R i L i

v R i L i

ω ω

ω

= + +

= − (4.3)

Fig. 4.2 is the block diagram of the overall open-loop PMSM drive system

designed in MATLAB/SIMULINK®. In this figure, transformations are in the following

order from left to right: abc reference frame to -stationary reference frame

transformation, -stationary reference frame to dq-rotor reference frame

transformation and finally dq-rotor reference frame to abc-frame transformation.

DQ

DQ

91

mean(VI.signals.values)/(80*max(ialoaded.signals.values)/sqrt(2))

For cos (Phi)

((mean(P.signals.values))/(24))/((3*80*max(ialoaded.signals.values)/sqrt(2))*(mean(VI.signals.values)/(80*max(ialoaded.signals.values)/sqrt(2))))For eff.

w m_rpm

-K-

w e_to_w mvc

vb

va

qdr2abc

ia

delta

abc2qd1

VI

Tem

Scope10

Product1

Product

f(u)

Pf=P/S

Pactive

PF

P

Load Torque1

PM-SM

idiq

Temwr_sliptheta_e

phi_dphi_q

wrdelta

F&P BPMmotor model

emVabc

IabcPQ

3-phase Instantaneous

Active & Reactive Pow er

vds

abc2qd0qds2qdr

vqs

2

vqr

1

vdr

u[1]*u[3] - u[2]*u[4]

qds2qr

u[2]*u[3] + u[1]*u[4]

qds2dr

(2*u[1]- u[2] - u[3])/3

abc2q

(u[3] -u[2])/sqrt(3)

abc2d

Mux

Mux1Mux

Mux

sin(u)

Fcn1

cos(u)

Fcn4

thetar

3

vc

2

vb

1

va

Fig. 4.2. Overall open-loop control block diagram of PMSM in SIMULINK®.

Fig. 4.3. Abc to stationary and stationary to dq rotor reference frame transformation

block.

where and or and are the stationary reference frame voltages converted

from the motors abc terminal voltages by using Clark’s Transformation which are given

by:

qsv dsv vα vβ

92

(2 )3

( )3

a b cqs

c bds

v v vv v

v vv v

α

β

− −= =

−= =

(4.4)

From Fig. 4.3, the stationary to rotor reference frame voltage equations

corresponding to the Park’s Transformation under the assumption that the rotor angle

representation is chosen between the rotor q-axis and the stationary q-axis, as shown in

Fig. 4.4, are given by:

cos sin

sin cos

s sq q r d

s sd q r d

v v v

v v v

θ θ

θ θ

= −

= +

r

r

(4.5)

The coordinate representation of the stationary reference frame is shown in

Fig. 4.4.

sq

sd

rd

rq

rθ

rθ

Fig. 4.4. Rotor reference frame and stationary reference frame coordinate representation.

93

qdr2qdsqds2abc

3

ic

2

ib

1

ia

-u[1]/2 +sqrt(3)*u[2]/2

qds2c

-u[1]/2 -sqrt(3)*u[2]/2

qds2b

u[1]

qds2au[1]*u[3] + u[2]*u[4]

qdr2qs

u[1]*u[4] - u[2]*u[3]

qdr2ds

Mux

Mux1

Mux

Mux

sin(u)

Fcn1

cos(u)

Fcn3

thetar

2

iq

1

id

Fig. 4.5. Dq rotor reference frame to stationary and stationary to abc transformation

block.

From Fig. 4.5, the rotor frame to stationary frame current equations

corresponding to the inverse Park’s Transformation, under the assumption that the rotor

angle representation is chosen as between the rotor q-axis and the stationary q-axis as

shown in Fig. 4.4, are given by:

cos sin

cos sin

sq q r d

sd d r q

i i i

i i i

θ θ

θ θ

= +

= −

r

r

(4.6)

Again from Fig. 4.5, the stationary frame to the abc frame current transformation

corresponding to the inverse Clark’s Transformation can be expressed as:

1 32 21 32 2

a qs

b qs

c qs

i i

i i

i i

=

=− −

=− +

ds

ds

i

i

(4.7)

From Fig. 4.2, the inside of the PMSM model is presented in Fig. 4.6.

94

wr

Vd

Vq

id

iq

wr_slip

electrical_pos.

9

delta

8

wr

7

out_7

6

out_6

5

out_5

4

out_4

3

out_3

2

out_2

1

out_1

0

wr=0 if wr<0

p/J

p/J

we

lPM1

Lamdam

lPM

Lamdam

lPM

1

RsTransfer Fcn1,Lq

1

RsTransfer Fcn,Ld

Switch

Sum5

Sum4

Sum3

Sum2

Sum1

Sum

Product2

Product1

Product

LqLq

Lq

Lq

-K-

Ld-Lq

Ld

Ld

1/s

Integrator1

1/s

Integrator

1/s

Integrator

-K-

1.5*p

3

in_3

2

in_2

1

in_1

Fig. 4.6. Steady state dq-axis rotor reference frame motor model.

Equations for Fig. 4.6 can be summarized as:

95

3 ( ( ) )2 2

( ( ) ) ( )2

q s q r d d r r

d s d r q q

em i q d q q d

r e rmem mech damp

v R i L i

v R i L i

PT i L L i i

d t tJT T T JP dt dt

ω ω λ

ω

λ

ω ω ω

= + +

= −

= + −

−+ − = =

(4.8)

where damping torque, , is assumed to be zero. dampT

0

0

( ) ( ) ( )

( ) (0) (0)

( ( ) ) ( )

( ) ( )2

r e

t

r e r e

r e r

t

r e em mech damp

t t t

t dt

d t d tdt dt

Pt T T TJ

δ θ θ

ω ω θ θ

ω ω ω

ω ω

= −

= − + −

−=

− = + −

∫

∫ dt

(4.9)

where is the initial value of the load angle, , also known as ‘delto’ in

the simulation.

(0) (0)r eθ θ− (0)δ

When the steady-state rotor speed is equal to the synchronous speed er t ωω =)( ,

δ (load angle) is constant and 0=rslipω ( 0)( =− er t ωω ). It is always better to get

0)0( =eθ . This is done by aligning the d-axis (synchronous reference frame) with the

a-axis or stationary reference frame as shown in Fig. 4.7.

96

δ

rθ eθ

rq

axisaqs

,

eq

−

eq sq

Fig. 4.7. Rotor ref. frame, synchronous ref. frame, and stationary ref. frame

representation as vectors.

For ease the -axis will also be aligned with the a-axis which makes the rotor

initial position zero (we need to be aware of this in the experiment also). If the

SIMULINK

rq

® open-loop control is used in the transient analysis, then the synchronous

speed, , is chosen to be zero because the initial speed of the rotor is zero in the

beginning of transient-state. In steady-state the desired speed is 2*pi*40 electrical rad/s

for 100 rpm (48 rotor poles). Additionally, (4.3) is used for designing the transient

PMSM model instead of (4.2).

eω

For the correct load angle in the SIMULINK® model, the ‘delto’ value in the

motor model should be equal to pi/2 for the following reason:

Let’s assume that the input voltage waveform is ))0(cos( eema tVV θω += (our

input is a sine wave), which means 0)0( =eθ . If )0(eθ is not zero (as with a sine wave

excitation of )sin( tVV ema ω= (like the model used in SIMULINK®) which equals

)2/cos( πω −tV em where 2/)0( πθ −=e ), the no-load steady-state value of δ will be

2/π− instead of zero, therefore ‘delto’ will be 2/π [10].

97

4.3. Power Factor and Efficiency Calculations in Open-Loop Matlab/Simulink®

Model

To verify that the simulated model is exactly like the real motor more

characterization tests should be conducted, such as power factor and efficiency

comparisons. These can be calculated in the MATLAB/SIMULINK® platform in two

ways. One way is by using SIMULINK® internal power measurement blocks. The other

way to obtain these is by using the MATLAB® workspace with the following equations:

cose _cos_

( . . )max( . . )80

2

rms rms

rms rms

rms rms

V IR al Power PPFApparent Power S V I

P mean VI signals valuesia signals valuesS V I

φφ= = = =

=⎛ ⎞⎟⎜= = ⎟⎜ ⎟⎟⎜⎝ ⎠

(4.10)

where is the abbreviation for Power Factor, is the apparent power, and 80 is the

rms value of the applied sinewave voltage.

PF S

P is calculated by taking the average (mean function) of VI which is the product

of (phase A voltage) and (phase A current). They are shown as the scope names in

SIMULINK

av ai® shown in Fig. 4.2. Below is the derivation of the efficiency formula which

is used in MATLAB®:

;3 cos 2

( )/ 2

3 cos

out em me m

in rms rms

eem

rms rms

P T PP V I

TP

V I

ωη ω

φ

ω

ηφ

= = =

=

ω

(4.11)

where φ is the power factor angle between the motor voltage and current.

The formulas below are used to calculate the power factor and efficiency. They

are used in MATLAB® workspace after the simulation ends:

Power Factor (PF):

mean(VI.signals.values)/(80*max(ia.signals.values)/sqrt(2))

98

Motor Efficiency (Eff.):

((mean(P.signals.values))/(24))/((3*80*max(ia.signals.values)/sqrt(2))*

(mean(VI.signals.values)/(80*max(ia.signals.values)/sqrt(2))))

In open-loop steady-state control there is a transient in the beginning, so to have

the real values data should be collected when transient passes, otherwise the transient

will dramatically change the PF and eff. values. To do that, in the scope parameters the

‘limit data points to last’ description should be sufficient enough to pass and not include

those transient effects. For example, 5000 points is chosen in the simulation and that was

enough for the calculation of PF and eff. Once the simulation is run the signal values

which are collected from the SIMULINK® scopes are run in the MATLAB® workspace

using the equations above so that the PF and eff. values are obtained. Similarly, the

SIMULINK® internal Power Measurement blocks can be used to calculate the PF and

Active (Real) Power easily. With those results, the eff. block can also be built in

SIMULINK®, as given below:

( )24

e

out

in active

P scopePP P

η= = (4.12)

where 24 is the pole pair number of the 48 pole PM spindle motor, is the

electrical output power of the motor and is the resultant active power part of the

output of the built-in internal power block. If better power results are desired then they

can be averaged out using ‘mean (P.signals.values)’, but it can only be done in the

MATLAB

( )eP scope

activeP

® command screen.

99

CHAPTER V

CONVENTIONAL DIRECT TORQUE CONTROL (DTC)

OPERATION OF PMSM DRIVE

5.1. Introduction and Literature Review

Today there are basically two types of instantaneous electromagnetic

torque-controlled AC drives used for high-performance applications: vector and direct

torque control (DTC) drives. The most popular method, vector control was introduced

more than 25 years ago in Germany by Hasse [18], Blaske [13], and Leonhard. The

vector control method, also called Field Oriented Control (FOC) transforms the motor

equations into a coordinate system that rotates in synchronism with the rotor flux vector.

Under a constant rotor flux amplitude there is a linear relationship between the control

variables and the torque. Transforming the AC motor equations into field coordinates

makes the FOC method resemble the decoupled torque production in a separately

excited DC motor. Over the years, FOC drives have achieved a high degree of maturity

in a wide range of applications. They have established a substantial world wide market

which continues to increase [6].

No later than 20 years ago, when there was still a trend toward standardization of

control systems based on the FOC method, direct torque control was introduced in Japan

by Takahashi and Nagochi [19] and also in Germany by Depenbrock [20], [21], [22].

Their innovative studies depart from the idea of coordinate transformation and the

analogy with DC motor control. These innovators proposed a method that relies on a

bang-bang control instead of a decoupling control which is the characteristic of vector

control. Their technique (bang-bang control) works very well with the on-off operation

of inverter semiconductor power devices.

100

After the innovation of the DTC method it has gained much momentum, but in

areas of research. So far only one form of a DTC of AC drive has been marketed by an

industrial company, but it is expected very soon that other manufacturers will come out

with their own DTC drive products [2].

The basic concept behind the DTC of AC drive, as its name implies, is to control

the electromagnetic torque and flux linkage directly and independently by the use of six

or eight voltage space vectors found in lookup tables. The possible eight voltage space

vectors used in DTC are shown in Fig. 5.1 [2].

D

Q

60

6 (101)V

1(100)V

2 (110)V3 (010)V

4 (011)V

5 (001)V

0 (000)V

7 (111)V

Fig. 5.1. Eight possible voltage space vectors obtained from VSI.

The typical DTC includes two hysteresis controllers, one for torque error

correction and one for flux linkage error correction. The hysteresis flux controller makes

the stator flux rotate in a circular fashion along the reference trajectory as shown in

Fig. 5.2. The hysteresis torque controller tries to keep the motor torque within a

pre-defined hysteresis band.

101

D

Q

2θ

1θ

6θ5V 6V

3V 2V

4V1V

1V

2V

3V

1V

2V

3V4

V3

V4V

5V

4V

5V

6V

5V

6V

6V

1V

2V

3θ

4θ

5θ

Fig. 5.2. Circular trajectory of stator flux linkage in the stationary DQ-plane.

At every sampling time the voltage vector selection block decides on one of the

six possible inverter switching states ( , , ) to be applied to the motor terminals.

The possible outputs of the hysteresis controller and the possible number of switching

states in the inverter are finite, so a look-up table can be constructed to choose the

appropriate switching state of the inverter. This selection is a result of both the outputs

of the hysteresis controllers and the sector of the stator flux vector in the circular

trajectory.

aS S Sb c

There are many advantages of direct torque control over other high-performance

torque control systems such as vector control. Some of these are summarized as follows:

• The only parameter that is required is stator resistance

102

• The switching commands of the inverter are derived from a look-up

table, simplifying the control system and also decreasing the processing

time unlike a PWM modulator used in vector control

• Instead of current control loops, stator flux linkage vector and torque

estimation are required so that simple hysteresis controllers are used for

torque and stator flux linkage control

• Vector transformation is not applied because stator quantities are enough

to calculate the torque and stator flux linkage as feedback quantities to be

compared with the reference values

• The rotor position, which is essential for torque control in a vector

control scheme, is not required in DTC (for induction and synchronous

reluctance motor DTC drives)

Once the initial position of the rotor magnetic flux problem is solved for PMSM

drives by some initial rotor position estimation techniques or by bringing the rotor to the

known position, DTC of the PMSM can be as attractive as DTC of an induction motor. It

is also easier to implement and as cost-effective (no position sensor is required) when

compared to vector controlled PMSM drives. The DTC scheme, as its name indicates, is

focused on the control of the torque and the stator flux linkage of the motor, therefore, a

faster torque response is achieved over vector control. Furthermore, due to the fact that

DTC does not need current controller, the time delay caused by the current loop is

eliminated.

Even though the DTC technique was originally proposed for the induction

machine drive in the late 1980’s, its concept has been extended to the other types of AC

machine drives recently, such as switched reluctance and synchronous reluctance

machines. In the late 90s, DTC techniques for the interior permanent magnet

synchronous machine appeared, as reported in [23], [24].

Although there are several advantages of the DTC scheme over vector control, it

still has a few drawbacks which are explained below:

103

• A major drawback of the DTC scheme is the high torque and stator flux

linkage ripples. Since the switching state of the inverter is updated once

every sampling time, the inverter keeps the same state until the outputs of

each hysteresis controller changes states. As a result, large ripples in

torque and stator flux linkage occur.

• The switching frequency varies with load torque, rotor speed and the

bandwidth of the two hysteresis controllers.

• Stator flux estimation is achieved by integrating the difference between

the input voltage and the voltage drop across the stator resistance (by the

back-EMF integration as given in (5.9)). The applied voltage on the

motor terminal can be obtained either by using a DC-link voltage sensor,

or two voltage sensors connected to the any two phases of the motor

terminals. For current sensing there should be two current sensors

connected on any two phases of the motor terminals. Offset in the

measurements of DC-link voltage and the stator currents might happen,

because for current and voltage sensing, however, temperature sensitive

devices, such as operational amplifiers, are normally used which can

introduce an unwanted DC offset. This offset may introduce large drifts

in the stator flux linkage computation (estimation) thus creating an error

in torque estimation (torque is proportional to the flux value) which can

make the system become unstable.

• The stator flux linkage estimation has a stator resistance, so any variation

in the stator resistance introduces error in the stator flux linkage

computation, especially at low frequencies. If the magnitude of the

applied voltage and back-EMF are low, then any change in the resistance

will greatly affect the integration of the back-EMF.

• Because of the constant energy provided from the permanent magnet on

the rotor the rotor position of motor will not necessarily be zero at start

up. To successfully start the motor under the DTC scheme from any

104

position (without locking the motor at a known position), the initial

position of the rotor magnetic flux must be known. Once it is started

properly, however, the complete DTC scheme does not explicitly require

a position sensor.

From the time the DTC scheme was discovered for AC motor drives, it was

always inferior to vector control because of the disadvantages associated with it. The

goal is to bring this technology as close to the performance level of vector control and

even exceed it while keeping its simple control strategy and cost-effectiveness. As a

result, many papers have been presented by several researchers to minimize or overcome

the drawbacks of the DTC scheme. Here are some of the works that have been done by

researchers to overcome the drawbacks for the most recent AC drive technology using

direct torque control:

• Recently, researchers have been working on the torque and flux ripple

reduction, and fixing the switching frequency of the DTC system, as

reported in [25]-[30]. Additionally, they came up with a multilevel

inverter solution in which there are more voltage space vectors available

to control the flux and torque. As a consequence, smoother torque can be

obtained, as reported in [28] and [29], but by doing so, more power

switches are required to achieve a lower ripple and an almost fixed

switching frequency, which increases the system cost and complexity. In

the literature, a modified DTC scheme with fixed switching frequency

and low torque and flux ripple was introduced in [27] and [30]. With this

design, however, two PI regulators are required to control the flux and

torque and they need to be tuned properly. Very recently Rahman [31]

proposed a method for torque and flux ripple reduction in interior

permanent magnet synchronous machines under an almost fixed

switching frequency without using any additional regulators. This method

is a modified version of the previously discovered method for the

induction machine by the authors in [32].

105

• Stator flux linkage estimation by the integration of the back-EMF should

be reset regularly to reduce the effect of the DC offset error. There has

been a few compensation techniques related to this phenomenon proposed

in the literature [33]-[35] and [16]. Chapuis et al. [18] introduced a

technique to eliminate the DC offset, but a constant level of DC offset is

assumed which is usually not the case. In papers [33]-[35] and [16],

low-pass filters (LPFs) have been introduced to estimate the stator flux

linkage. In [33], a programmable cascaded LPF was proposed instead of

the single-stage integrator to help decrease the DC offset error more than

the single-stage integrator for induction motor drives. More recently,

Rahman [36] has reached an approach like [33] with further investigation

and implementation for the compensation of DC offset error in a direct

controlled interior permanent magnet (IPM) synchronous motor drive. It

has been claimed and proven with simulation and experimental results

that programmable cascaded LPFs can also be adopted to replace the

single-stage integrator and compensate for the effect of DC offsets in a

direct-torque-controlled IPM synchronous motor drive, improving the

performance of the drive.

• The voltage drop in the stator resistance is very large when the motor runs

at low frequency such that any small deviations in stator resistance from

the one used in the estimation of the stator flux linkage creates large

errors between the reference and actual stator flux linkage vector. This

also affects the torque estimation as well. Due to these errors, the drive

can easily go unstable when operating at low speeds. The worst case

scenario might happen at low speed under a very high load. A handful of

researchers have recently pointed to the issue of stator resistance variation

for the induction machine. For example, fuzzy and proportional-integral

(PI) stator resistance estimators have been developed and compared for a

DTC induction machine based on the error between the reference current

106

and the actual one by Mir et al. [37]. On the other hand, they did not show

any detail on how to obtain the reference current for the stator resistance

estimation. Additionally, some stability problems of the fuzzy estimator

were observed when the torque reference value was small. As reported in

[38], fuzzy logic based stator resistance observers are introduced for

induction motor. Even though it is an open-loop controller based on fuzzy

rules, the accuracy of estimating the stator resistance is about 5% and

many fuzzy rules are necessary. This resulted in having to conduct

handful numbers of extensive experiments to create the fuzzy rules

resulting in difficulty in implementation. Lee and Krishnan [39]

contributed a work related to the stator resistance estimation of the DTC

induction motor drive by a PI regulator. An instability issue caused by the

stator estimation error in the stator resistance, the mathematical

relationships between stator current, torque and flux commands, and the

machine parameters are also analyzed in their work. The stator

configuration of all AC machines is almost the same, so the stator

resistance variation problem still exists for permanent magnet

synchronous motors. Rahman et al. [40] reported a method, for stator

resistance estimation by PI regulation based on the error in flux linkage. It

is claimed that any variation in the stator resistance of the PM

synchronous machine will cause a change in the amplitude of the actual

flux linkage. A PI controller works in parallel with the hysteresis flux

controller of the DTC such that it tracks the stator resistance by

eliminating the error in the command and the actual flux linkage. One

problem with this method was that the rotor position was necessary to

calculate the flux linkage. Later on the same author proposed a similar

method but this time the PI stator resistance estimator was able to track

the change of the stator resistance without requiring any position

information.

107

• The back-EMF integration for the stator flux linkage calculation, which

runs continuously, requires a knowledge of the initial stator flux position,

0tsλ =, at start up. In order to start the motor without going in the wrong

direction, assuming the stator current is zero at the start, only the rotor

magnetic flux linkage should be considered as an initial flux linkage

value in the integration formula. The next step is to find its position in the

circular trajectory. The initial position of the rotor is not desired to be

sensed by position sensors due to their cost and bulky characteristics,

therefore some sort of initial position sensing methods are required for

permanent magnet synchronous motor DTC applications. A number of

works, [41]-[52], have been proposed recently for the detection of the

initial rotor position estimation at standstill for different types of PM

motors. Common problems of these methods include: most of them fail at

standstill because the rotor magnet does not induce any voltage, so no

information of the magnetization is available; position estimation is load

dependent; excessive computation and hardware are required; instead of a

simple voltage vector selection method used in the DTC scheme, those

estimation techniques need one or more pulse width-modulation (PWM)

current controllers. Recently, a better solution was introduced for the

rotor position estimation. It is accomplished by applying high-frequency

voltage to the motor, as reported in [50]-[52]. This approach is adapted to

the DTC of interior permanent magnet motors for initial position

estimation by Rahman et. al. [36].

5.2. Principles of Conventional DTC of PMSM Drive

The basic idea of direct torque control is to choose the appropriate stator voltage

vector out of eight possible inverter states (according to the difference between the

108

reference and actual torque and flux linkage) so that the stator flux linkage vector rotates

along the stator reference frame (DQ frame) trajectory and produces the desired torque.

The torque control strategy in the direct torque control of a PM synchronous motor is

explained in Section 5.2.1. The flux control is discussed following the torque control

section.

5.2.1. Torque Control Strategy in DTC of PMSM Drive

Before going through the control principles of DTC for PMSMs, an expression

for the torque as a function of the stator and rotor flux will be developed. The torque

equation used for DTC of PMSM drives can be derived from the phasor diagram of

conventional or permanent magnet synchronous motor shown in Fig. 5.3.

δ

d-axis

q-axis

s sI R

E

s sjI X

sV

sIϕ

rλ

Fig. 5.3. Phasor diagram of a non-saliet pole synchronous machine in the motoring

mode.

109

R ss LjjX ω=

°∠0V °∠δfE

Fig. 5.4. Electrical circuit diagram of a non-salient synchronous machine at constant

frequency (speed).

When the machine is loaded through the shaft, the motor will take real power.

The rotor will then fall behind the stator rotating field. From the circuit diagram, shown

in Fig. 5.4, the motor current expression can be written as:

0 0s ss

s s s

V E V EIR jX Z

δϕ

∠ − ∠ ∠ − ∠= =

+ ∠δ (5.1)

where 2 2s s sZ R X= + , also s e sX Lω=

and 1tan s

s

XR

ϕ −⎛ ⎞⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜⎝ ⎠

Assuming a reasonable speed such that the sX term is higher than the resistance, sR ,

such that sR can be neglected, then s sZ X≈ and 2π

ϕ≈ . sI can then be rewritten as:

0 2ss

s s

EVIX X

πδ∠ −∠

= − (5.2)

Such that the real part of sI is:

110

Re[ ] cos cos cos2 2

cos sin2

ss s

s s

s s

V EI IX X

E EX X

π πϕ δ

πδ δ

⎛ ⎞ ⎛⎟ ⎟⎜ ⎜= = − − −⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎝ ⎠ ⎝

⎛ ⎞⎟⎜=− − =−⎟⎜ ⎟⎜⎝ ⎠

⎞⎠

(5.3)

the developed power is given by:

3 Re[ ] 3 cosi s s s sP V I V I ϕ= = (5.4)

Substituting (5.3) into (5.4) yields:

3 sinsi

s

V EPX

δ=− Watts/phase (5.5)

This power is positive when δ negative, meaning that when the rotor field lags

the stator field the machine is operating in the motoring region. When the machine

is operating in the generation region. The negative sign in (5.5) can be dropped,

assuming that for motoring operation a negative δ is implied.

0δ>

If the losses of the machine are ignored, the power can be expressed as the

shaft (output) power as well:

iP

2i o e eP P T

Pω= = m (5.6)

When combining (5.5) and (5.6), the magnitude of the developed torque for a

non-salient synchronous motor (or surface-mounted permanent magnet synchronous

motor) can be expressed as:

3 s2

3 sin2

eme s

s

PTX

PL

δω

δ

⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜⎝ ⎠

⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜⎝ ⎠

s

s r

V E

λ λ

in (5.7)

111

where is the torque angle between flux vectors δ sλ and . If the rotor flux remains

constant and the stator flux is changed incrementally by the stator voltage,

rλ

sV , then the

torque variation, , expression can be written as: emTΔ

32em

s

PTL

δ⎛ ⎞ +⎟⎜Δ = Δ⎟⎜ ⎟⎜⎝ ⎠

s s rλ Δλ λsin (5.8)

where the bold terms in the above expressions indicate vectors.

As it can be seen from (5.8), if the load angle, , is increased then torque

variation is increased. To increase the load angle, δ , the stator flux vector should turn

faster than rotor flux vector. The rotor flux rotation depends on the mechanical speed of

the rotor, so to decrease load angle, δ , the stator flux should turn slower than rotor flux.

Therefore, according to the torque (5.7), the electromagnetic torque can be controlled

effectively by controlling the amplitude and rotational speed of stator flux vector,

δ

sλ . To

achieve the above phenomenon, appropriate voltage vectors are applied to the motor

terminals. For counter-clockwise operation, if the actual torque is smaller than the

reference value, then the voltage vectors that keep the stator flux vector, sλ , rotating in

the same direction are selected. When the load angle, δ , between sλ and increases

the actual torque increases as well. Once the actual torque is greater than the reference

value, the voltage vectors that keep stator flux vector,

rλ

sλ , rotating in the reverse

direction are selected instead of the zero voltage vectors. At the same time, the load

angle, , decreases thus the torque decreases. The reason the zero voltage vector is not

chosen in the DTC of PMSM drives will be discussed later in this chapter. A more

detailed look at the selection of the voltage vectors and their effect on torque and flux

results will be discussed later as well. Referring back to the discussion above, however,

torque is controlled via the stator flux rotation speed, as shown in Fig. 5.5. If the speed

of the stator flux is high then faster torque response is achieved.

δ

112

reωsω

δ

sλ

rλ

Im

Rereθsθ

Fig. 5.5. Rotor and stator flux linkage space vectors (rotor flux lagging stator flux) [35].

5.2.2. Flux Control Strategy in DTC of PMSM Drive

If the resistance term in the stator flux estimation algorithm is neglected, the

variation of the stator flux linkage (incremental flux expression vector) will only depend

on the applied voltage vector as shown in Fig. 5.6 [53].

D

Q

2θ

1θ

6θ

3θ

4θ

5θ

0sλ

sλ∗

2Hλ

3V

2V

4V

3V4V

5V4V

5V

6V

5V

6V

1V6V

1V

2V

2V

2V3V

4V

5V 6V

1V

sλ2 sV T

3V

Fig. 5.6. Incremental stator flux linkage space vector representation in the DQ-plane.

113

For a short interval of time, namely the sampling time, , the stator flux

linkage,

sT =Δt

sλ , position and amplitude can be changed incrementally by applying the stator

voltage vector, sV . As discussed above, the position change of the stator flux linkage

vector, sλ

)

, will affect the torque. The stator flux linkage of a PMSM that is depicted in

the stationary reference frame is written as:

(s s s sR dt= −∫λ V i (5.9)

During the sampling interval time or switching interval, one out of the six

voltage vectors is applied, and each voltage vector applied during the pre-defined

sampling interval is constant, s (5.9) can be rewritten as:

s s s s s t=0t - R dt + λ= ∫λ V i (5.10)

where s t=0λ is the initial stator flux linkage at the instant of switching, Vs is the

measured stator voltage, is , is the measured stator current, and sR is the estimated stator

resistance. When the stator term in stator flux estimation is removed implying that the

end of the stator flux vector, sλ , will move in the direction of the applied voltage vector,

as shown in Fig. 5.6, we obtain:

( )V λs sddt

= (5.11)

or

Δts sΔλ =V (5.12)

The goal of controlling the flux in DTC is to keep its amplitude within a

pre-defined hysteresis band. By applying a required voltage vector stator flux linkage

amplitude can be controlled. To select the voltage vectors for controlling the amplitude

of the stator flux linkage the voltage plane is divided into six regions, as shown in

Fig. 5.2.

114

In each region two adjacent voltage vectors, which give the minimum switching

frequency, are selected to increase or decrease the amplitude of stator flux linkage,

respectively. For example, according to the Table III, when the voltage vector is

applied in Sector 1, then the amplitude of the stator flux increases when the flux vector

rotates counter-clockwise. If is selected then stator flux linkage amplitude decreases.

The stator flux incremental vectors corresponding to each of the six inverter voltage

vectors are shown in Fig. 5.1.

2V

3V

sω

s sTω

sλ

0sλ

Im

Re0sθ

sθ

s sV T

Direct (Flux) component

Indirect (Torque) component

Fig. 5.7. Representation of direct and indirect components of the stator flux linkage

vector [35].

Fig. 5.7 is a basic graph that shows how flux and torque can be changed as a

function of the applied voltage vector. According to the figure, the direct component of

applied voltage vector changes the amplitude of the stator flux linkage and the indirect

component changes the flux rotation speed which changes the torque. If the torque needs

to be changed abruptly then the flux does as well, so the closest voltage vector to the

indirect component vector is applied. If torque change is not required, but flux amplitude

is increased or decreased then the voltage vector closest to the direct component vector

is chosen. Consequently, if both torque and flux are required to change then the

appropriate resultant mid-way voltage vector between the indirect and direct components

is applied [35]. It seems obvious from (5.9) that the stator flux linkage vector will stay at

115

its original position when zero voltage vectors and are applied. This is

true for an induction motor since the stator flux linkage is uniquely determined by the

stator voltage. On the other hand, in the DTC of a PMSM, the situation of applying the

zero voltage vectors is not the same as in induction motors. This is because the stator

flux linkage vector will change even when the zero voltage vectors are selected since the

magnets rotate with the rotor. As a result, the zero voltage vectors are not used for

controlling the stator flux linkage vector in a PMSM. In other words, the stator flux

linkage should always be in motion with respect to the rotor flux linkage vector [24].

(000)S (111)Sa a

5.2.3. Voltage Vector Selection in DTC of PMSM Drive

As discussed before, the stator flux is controlled by properly selected voltage

vectors, and as a result the torque by stator flux rotation. The higher the stator vector

rotation speed the faster torque response is achieved.

The estimation of the stator flux linkage components described previously

requires the stator terminal voltages. In a DTC scheme it is possible to reconstruct those

voltages from the DC-link voltage, , and the switching states ( , , ) of a

six-step voltage-source inverter (VSI) rather than monitoring them from the motor

terminals. The primary voltage vector,

dcV aS bS cS

sv , is defined by the following equation:

(2 /3) (4 /3)2 (3

j js a b cv v e v eπ= + +v )π (5.13)

where , , and are the instantaneous values of the primary line-to-neutral voltages.

When the primary windings are fed by an inverter, as shown in Fig. 5.8, the primary

voltages , and are determined by the status of the three switches, , , and

. If the switch is at state 0 that means the phase is connected to the negative and if it is

at 1 it means that the phase is connected to the positive leg.

av bv cv

av bv cv aS bS

cS

116

aS bS cS1

0

1

0

1

0

dcV

D

Q

Fig. 5.8. Voltage source inverter (VSI) connected to the R-L load [19].

For example, is connected to if is one, otherwise is connected to zero. This

is similar for and . The voltage vectors that are obtained this way are shown in

Fig. 5.1. There are six nonzero voltage vectors: , , …, and and

two zero voltage vectors: and . The six nonzero voltage vectors are

apart from each other as in Fig. 5.1.

av dcV aS av

bv cv

1(100)V 2 (110)V 6 (101)V

7 (000)V 8 (111)V 60

The stator voltage space vector (expressed in the stationary reference frame)

representing the eight voltage vectors can be shown by using the switching states and the

DC-link voltage, , as: dcV

(2 /3) (4 /3)2( , , ) ( )3

j js a b c dc a b cS S S V S S e S eπ= + +v π

)

a ab cav v v= − ( ) / 3b bc abv v v= −

(5.14)

where is the DC-link voltage and the coefficient of 2/3 is the coefficient comes from

the Park’s Transformation. Equation (5.14) can be derived by using the line-to-line

voltages of the AC motor which can be expressed as: ,

, and . The stator phase voltages (line-to-neutral

voltages) are required for (5.14). They can be obtained from the line-to-line voltages as

, , and . If the line-to-line voltages

dcV

( )ab dc a bv V S S= −

(bc dc b cv V S S= − ( )ca dc c av V S S= −

( ) / 3 ( ) / 3c ca bcv v v= −

117

in terms of the DC-link voltage, , and switching states are substituted into the stator

phase voltages it gives:

dcV

1 (2 )3a dc a b cv V S S S= − −

1 ( 23b dc a bv V S S S= − + − )c (5.15)

1 ( 23c dc a bv V S S S= − − + )c

Equation (5.15) can be summarized by combining with (5.13) as:

1Re( ) (2 )3a s dc a b cv V S S= = − −v S

1Re( ) ( 2 )3b s dc a bv V S S= = − + −v cS (5.16)

1Re( ) ( 2 )3c s dc a bv V S S= = − − +v cS

To determine the proper applied voltage vectors, information from the torque and

flux hysteresis outputs, as well as stator flux vector position, are used so that circular

stator flux vector trajectory is divided into six symmetrical sections according to the non

zero voltage vectors as shown in Fig. 5.2.

iV

1iV +2iV +

1iV −2iV −

3iV +

sλ

D

Q

2θ

1θ

6θ

emTsλ

emTsλ

emTsλ

emTsλ

Fig. 5.9. Voltage vector selection when the stator flux vector is located in sector i [35].

118

According to Fig. 5.9, while the stator flux vector is situated in sector i, voltage

vectors and V have positive direct components, increasing the stator flux

amplitude, and V and V have negative direct components, decreasing the stator flux

amplitude. Moreover, V and V have positive indirect components, increasing the

torque response, and V and V have negative indirect components, decreasing the

torque response. In other words, applying V increases both torque and flux but

applying V increases torque and decreases flux amplitude [35].

i+1V i-1

i+2 i-2

i+1 i+2

i-1 i-2

i+1

i+1

The switching table for controlling both the amplitude and rotating direction of

the stator flux linkage is given in Table III.

TABLE III

SWITCHING TABLE

V2(110) V3(010) V4(001) V5(101) V6(110) V1(110)V6(101) V1(100) V2(010) V3(011) V4(110) V5(110)V3(010) V4(011) V5(101) V6(100) V1(110) V2(110)V5(001) V6(101) V1(110) V2(010) V3(110) V4(110)

θθ(1) θ(2) θ(3) θ(4) θ(5) θ(6)

ϕ τ

1ϕ=

0ϕ=

1τ =0τ =1τ =0τ =

The voltage vector plane is divided into six sectors so that each voltage vector

divides each region into two equal parts. In each sector, four of the six non-zero voltage

vectors may be used. Zero vectors are also allowed. All the possibilities can be tabulated

into a switching table. The switching table presented by Rahman et al [24] is shown in

Table III. The output of the torque hysteresis comparator is denoted as τ , the output of

the flux hysteresis comparator as ϕ and the flux linkage sector is denoted as θ . The

torque hysteresis comparator is a two valued comparator; means that the actual

value of the torque is above the reference and out of the hysteresis limit and

0τ =

=1τ means

that the actual value is below the reference and out of the hysteresis limit. The flux

hysteresis comparator is a two valued comparator as well where 1ϕ =

0ϕ=

means that the

actual value of the flux linkage is below the reference and out of the hysteresis limit and

means that the actual value of the flux linkage is above the reference and out of

119

the hysteresis limit. Rahman et al [24] have suggested that no zero vectors should be

used with a PMSM. Instead, a non zero vector which decreases the absolute value of the

torque is used. Their argument was that the application of a zero vector would make the

change in torque subject to the rotor mechanical time constant which may be rather long

compared to the electrical time constants of the system. This results in a slow change of

the torque. This reasoning does not make sense, since in the original switching table the

zero vectors are used when the torque is inside the torque hysteresis (i.e. when the torque

is wanted to be kept as constant as possible). This indicates that the zero vector must be

used. If the torque ripple needs to be kept as small as with the original switching table, a

higher switching frequency must be used if the suggestion of [24] is obeyed [6].

We define ϕ and to be the outputs of the hysteresis controllers for flux and

torque, respectively, and as the sector numbers to be used in defining the

stator flux linkage positions. In Table III, if , then the actual flux linkage is smaller

than the reference value. On the other hand, if , then the actual flux linkage is

greater than the reference value. The same is true for the torque.

τ

(1) (6)θ θ−

1=ϕ

0=ϕ

120

5.3. Control Strategy of DTC of PMSM

+-

+- 2θ

1θ

6θ

3θ

4θ

5θ

30

Voltage Source Inverter (VSI)

( )32em D D Q QΤ P λ i λ i= −

( )

( )0

0

D D s D D

Q Q s Q Q

V R i dt

V R i dt

λ λ

λ λ

= − +

= − +

∫∫

2 2s D Qλ λλ = +

arctan Qs

D

λθ

λ=

CurrentSensors

PMSM

Voltage VectorSelection Table

HysteresisControllers

A

B

C

1 ( 2 )3

D a

Q a b

i i

i i i

=

= +1 ( 2 )3

D a

Q a b

i i

i i i

=

= +,D QV V

Dc-link voltage

sλ

emT

emT ∗

sλ∗

sθ

Initial rotorposition

Stator flux and torque estimator

iV

1

0

1

0

0Dλ 0Qλ,

Fig. 5.10. Basic block diagram of DTC of PMSM.

Fig. 5.10 illustrates the schematic of the basic DTC controller for PMSM drives.

The command stator flux, *sλ , and torque, , magnitudes are compared with their

respective estimated values. The errors are then processed through the two hysteresis

comparators, one for flux and one for torque which operate independently of each other.

The flux and torque controller are two-level comparators. The digital outputs of the flux

controller have following logic:

*emT

1λd = for *s s Hλλ λ< − (5.17)

0λd = for *s s Hλλ λ< + (5.18)

where is the total hysteresis-band width of the flux comparator, and is the

digital output of the flux comparator.

λ2H λd

By applying the appropriate voltage vectors the actual flux vector, sλ , is

constrained within the hysteresis band and it tracks the command flux, *sλ , in a zigzag

121

path without exceeding the total hysteresis-band width. The torque controller has also

two levels for the digital output, which have the following logic:

emTd = 1 for (5.19) em

*em em TT <T + H

emTd = 0 for (5.20) *emem em TT T H< −

where is the total hysteresis-band width of the torque comparator, and is the

digital output of the torque comparator.

emT2HemTd

0

1 112 23 30

2 21 1 12 2 2

D a

Q

c

b

f ff f

ff

⎡ ⎤⎢ ⎥− −⎢ ⎥⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢= − −⎢ ⎥ ⎥

⎢ ⎥⎢⎢ ⎥ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎢ ⎥

⎢ ⎥⎢ ⎥⎣ ⎦

(5.21)

Knowing the output of these comparators and the sector of the stator flux vector,

the look-up table can be built such that it applies the appropriate voltage vectors via the

inverter in a way to force the two variables to predefined trajectories. If the switching

states of the inverter, the DC-link voltage of the inverter and two of the motor currents

are known then the stator voltage and current vectors of the motor in the DQ stationary

frame are obtained easily by a simple transformation. This transformation is called the

Clarke transformation [19] (5.21) as shown in Fig. 5.10. The DQ frame voltage and

current information can then be used to estimate the corresponding D-and Q-axes stator

flux linkages, Dλ and Qλ , which are given by:

( ) ( 1) ( 1) ( )D D D s Dλ k λ k v k R i k= − + − − sT (5.22)

( ) ( 1) ( 1) ( )Q Q Q s Q sλ k λ k v k R i k= − + − − T (5.23)

where and are present and previous sampling instants, respectively, k 1k− Dv and

are the stator voltages in DQ stationary reference frame,

Qv

( ) ( ( 1) ( )) / 2D D Di k i k i k= − +

122

and ( ) ( ( 1) ( )) / 2Q Q Qi k i k i k= − + are the average values of stator currents, Di and ,

derived from the present, , and previous, , sampling interval values of

the stator currents,

Qi

( )DQi k ( 1DQi k − )

sR is the stator resistance, and sT is the sampling time. The stator

flux linkage vector can be written as:

2 2 1 ( )( ) ( ) ( ) tan

( )Q

D QD

λ kk λ k λ k

λ k−⎛ ⎞⎟⎜ ⎟= + ∠ ⎜ ⎟⎜ ⎟⎜⎝ ⎠

sλ (5.24)

where 2( ) ( )D Q2λ k λ k+ is the magnitude of the stator flux linkage vector and

1 ( )tan

( )Q

D

λ kλ k

−⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

is the angle of stator flux linkage vector with respect to the stationary

D-axis in DQ frame (or a-axis in abc frame). The developed stationary DQ reference

frame electromagnetic torque in terms of the DQ frame stator flux linkages and currents

is given by

3( ) ( ) ( ) ( ) ( )2em Q D D QΤ k P λ k i k λ k i k= −

)

(5.25)

where is the number of pole pairs. P

As it can be seen form (5.22) and (5.23) the stator resistance is the only machine

parameter to be known in the flux, and consequently torque, estimation. Even though the

stator is the direct parameter seen in (5.22) and (5.23), there is an indirect (hidden) motor

parameter for DTC of PMSM drives. This parameter is the rotor flux magnitude which

constructs the initial values of the D- and Q-axes stator fluxes. If the rotor flux vector,

, is assumed to be aligned with the D-axis of the stationary reference frame, then

equals the rotor flux amplitude

rλ

( 1Dλ k− r2λ . If the rotor magnetic flux, , resides on

the D-axis (the rotor magnetic flux can be intentionally brought to the known position by

applying the appropriate voltage vector for a certain amount of time), then the initial

value of the Q-axis flux, , is considered to be zero, therefore there will not be

any initial starting problem for the motor. On the other hand, if the rotor is in a position

rλ

( 1Qλ k − )

123

other than the zero reference degree then both the and values should

be known to start the motor properly in the correct direction without oscillation.

Moreover, if the initial values of the DQ frame integrators are not estimated correctly

then those incorrect initial flux values will be seen as DC components in the integration

calculations of the DQ frame fluxes. This will cause the stator flux linkage space vector

to drift away from the origin centered circular path and if they are not corrected quickly

while motor is running then instability in the system will result quickly.

( 1)Dλ k− ( 1Qλ k − )

124

CHAPTER VI

PROPOSED DIRECT TORQUE CONTROL (DTC) OPERATION OF PMSM

DRIVE

6.1. Introduction and Principles of the Proposed DTC Drive

In conventional DTC drives, there are some drawbacks, such as the DC drifts in

the voltage and current measurements, wrong DQ-frame initial flux linkages being used

or having resistance term exists in the estimation algorithm of the DQ stationary

reference frame flux linkages [36]. These disadvantages of the basic DTC scheme are

claimed to be eliminated by the proposed DTC technique.

Unlike the conventional DTC scheme which has a voltage model [24], the

proposed method is based on the current model which is considered in the rotor

coordinate system (rotor reference frame). By using the current model, the stator

quantities need to be transferred to the rotor reference frame quantities (this is ensured

by the position information). In this thesis, position information is provided by three

cheap hall-effect sensors (1$<) instead of bulky and expensive encoders or resolvers.

One hall-effect sensor is mounted on the stator starting from the boundary of Sector 1

and 2, and the remaining two hall-effect sensors are placed 120 electrical degrees apart

from each other in the counter-clockwise direction as shown in Fig. 6.1. The sector

change of the stator flux linkage space vector on the trajectory can be easily determined

by the help of the six hall-effect sensor signals that accompany one complete electrical

cycle.

125

D

Q

2θ

1θ

6θ

3θ

4θ

5θ

2V3V

4V

5V 6V

1V30

aHbH

cH

bH ′aH ′

cH ′

Hall-1Hall-2

Hall-3

Fig. 6.1. Hall-effect sensor placement shown in DQ stationary reference frame.

The rotor reference frame stator flux linkages, and , (dq rotor reference

frame) from the current model and torque equation are the key formulas in the proposed

DTC drive shown in (2.21) and (2.44), respectively. Unlike conventional DTC drives

which use stator reference frame currents, currents in (2.21) and (2.44) are represented in

the rotor reference frame as discussed in Chapter II. In this case, continuous rotor

position information is needed for use in the Park’s Transformation because the Park’s

transformation is originally used in sinusoidal machines which need high resolution

position feedback for the complex vector control algorithm. High-resolution position

information can be obtained from an encoder or resolver generating high number of

pulses, but in this thesis three hall-effect sensors assisted by a simplified observer are

used to mimic the behavior of those expensive high-resolution position sensors. Usually

hall-effect sensors are used in BLDC motor drives that have trapezoidal back-EMF. In

BLDC motor drives, each phase current conducts 120 electrical degrees and their shapes

dλ qλ

126

are like square waveforms and their power devices are switched every 60 electrical

degrees to commutate the phase currents. The position sensor, therefore, has a resolution

of only 60 electrical degrees, thus an inexpensive low-resolution position sensor, such as

a hall-effect sensor, is suitable. On the other hand, in sinusoidal machines like PMSMs,

high-performance current vector control cannot be achieved and the operating

performance is inferior to the drive system which has low-resolution feedback position

information. The detailed explanation of position and speed sensing used in the proposed

DTC scheme by a simple observer assisted with three low-cost hall-effect sensors will be

discussed in the next section.

Once the position is estimated by the three hall-effect sensors, the Park’s

Transformation is applied to the three phase abc frame currents that are measured to get

the dq rotor reference frame currents. After that, the obtained dq rotor reference frame

currents are placed in (2.21) to estimate the stator dq axes flux linkages. With this

information, the stator flux linkage amplitude can be derived as:

2 2s d qλ λ λ= + Equation Section 6(6.1)

The stator flux linkage amplitude calculated from rotor reference frame

quantities is exactly the same as the one found by using the stator reference frame fluxes

Dλ and as shown in Fig. 6.2. Qλ

127

reω

sω

δ

sλ

rλ

Q

Dreθsθ

d dL i

q qL i

Dλ

Qλq

d

Fig. 6.2. Representation of the rotor and stator flux linkages both in the synchronous dq

and stationary DQ-planes.

Although the stator flux linkage amplitude calculation in both the stator and rotor

reference frames generates the same result, the exact position of the stator flux vector in

the DQ frame cannot be applied to the rotor reference frame flux components by just

using (5.27). The angle between the rotor reference frame q-axis flux and d-axis flux,

which is a constant value due to the DC values of the synchronous frame currents equals

to the load angle,δ , and the stator flux vector angle, sθ , with respect to the D-axis, is

given by:

arctan qs r r

d

λθ θ θ

λ

⎛ ⎞⎟⎜ ⎟= + = +⎜ ⎟⎜ ⎟⎜⎝ ⎠δ (6.2)

With the proposed method, the initial flux errors which cause drift in the basic

DTC scheme will not be added to the next calculations (integration). The only problem

in flux calculations might happen at high current and high temperature. The rotor

reference frame stator flux linkages possess d- and q-axes inductances along with a rotor

magnetic flux linkage amplitude which is affected by an increase in temperature. Under

magnetic saturation the q-axis inductance of an IPMSM varies and the d-axis inductance

is affected by the armature reaction therefore the saturation model for various load

128

conditions should be developed such that inductance should be expressed taking into

account magnetic saturation. Also, when the load is increased the amplitude of the rotor

magnetic flux linkage is increased (due to temperature rise) therefore it is necessary to

consider the change in the synchronous inductance ( s d qL L L= = in surface-mounted

PMSMs) because it is greatly affected due to magnetic saturation. Magnetic saturation

and temperature effects are not included in the proposed system, but it should be

considered and compensated for if there is high current and the operating points of the

current reside in the saturation region of the motor.

Basic DTC is based on a voltage model which is plagued by the stator resistance

error and the integration drift at low frequencies. On the other hand, the current model is

affected by parameter detuning (due to magnetic saturation) and also by the position

error, but it works pretty well from zero speed if a high-resolution position sensor such

as an encoder is used. In the literature, a combination of the voltage and current model is

proposed with a PI compensator such that the current model is predominant at low

speeds while the voltage model takes over at high speeds. Applications of this combined

method can be very costly due to the additional sensor (position). To make the overall

system cheaper, the current model is suggested instead of the voltage model therefore

only a very cheap position sensor is needed (hall-effect sensor). A complete block

diagram of the proposed DTC scheme is given in Fig. 6.3. Because the motor used in

this thesis is a high torque motor and in washing machine applications there is a certain

limit for the capacity of the load (approximately known load torque) in the drum such

that currents cannot bring the motor into the saturation region easily, thus inductance

compensation can be ignored. In certain applications where very high currents bring the

motor to the saturation region and temperature increases inside the motor are evident, the

inductance and amplitude of the rotor flux linkage should be compensated.

129

+-

+-

Vol

tage

Sou

rce

Inve

rter (

VSI

)

Cur

rent

Sens

ors

PMSM

Vol

tage

Vec

tor

Sele

ctio

n Ta

ble

Hys

tere

sis

Con

trolle

rs

A B C

Park

Tran

sfor

mat

ion

Stat

or fl

ux a

nd to

rque

est

imat

or in

roto

r ref

eren

ce fr

ame

Spee

dC

ontro

ller

(PI)

+-

Hal

l-Eff

ect

Sens

ors

1

0

1

0

Fig.

6.3

. Blo

ck d

iagr

am o

f the

pro

pose

d D

TC o

f PM

SM.

130

6.2. Position and Speed Estimation Method with Low-Resolution Position Sensor

In permanent magnet synchronous motors (PMSMs) and for every other

sinusoidal current controlled AC drive systems, in order to produce the sinusoidal

current commands and control the current phase accurately, high resolution position

information is essential. A high resolution position sensor such as an optical encoder

with large pulses per revolution or a resolver is therefore usually used.

Since the current model is used for the proposed DTC of a PMSM drive,

continuous rotor position information should be obtained. In this thesis, a low resolution

position sensor which has an accuracy of only 60 electrical degrees is used instead of a

high resolution sensor. Such low-resolution position feedback can be achieved by the

processed signals from three cost-effective hall-effect sensors as shown in Fig. 6.1. One

electrical cycle is divided into six intervals (representing the six sectors for the DTC

scheme) of 60 electrical degrees by the processed signals , , and . In this case,

the absolute position with an accuracy of 30 electrical degrees (mid point of two

consecutive hall-effect signals are chosen as a reference point) can be obtained, and as a

result reverse rotation at starting never occurs. The hall-effect sensor has advantages

such as smaller size and lower cost compared to an optical encoder or resolver.

However, since the position updates occur at every 60 electrical degree interval, the

necessary continuous position information for PMSM drives cannot be achieved by

using just three hall-effect sensors like in BLDC drives. Knowing this, vector

transformations (Park’s and inverse Park’s Transformations) and sinusoidal current

drives cannot be achieved by using low-resolution position sensing. The main goal of

this thesis is to estimate higher resolution position information from the rough position

information obtained from the three hall-effect sensors. The method of obtaining the

high-resolution position information from the three hall-effect sensor signals is

explained using Fig. 6.4 as follows:

aH bH cH

±

In Fig. 6.4, the actual positions, ( )real jθ , corresponding to , , 120 , 180 ,

, and 30 are obtained from the three hall-effect sensors at every 60 electrical

degrees. The index pointer of the corresponding component is defined as

0 60

240 0

j and 1j− is

131

the index pointer mapping the value of the corresponding component which is 60

electrical degrees behind the index j . The same convention is applied in the reverse way

for the index value 1j + , etc. For example, if a reference for j is chosen as , then 0

1j− represents . The time between two consecutive hall-effect signals, ,

which corresponds to the time between the previous hall-effect sensor position,

, and the next position,

300 ( )hallT j

( 1real jθ − ) ( )real jθ is depicted in Fig. 6.4. The average (midpoint)

speed, 1 2

tω−

, between and ( 1)real jθ − ( )real jθ can be calculated as follows [54]:

1 2

/ 3( )t

hallT jπ

ω−

= in electrical rad/sec (6.3)

. . . . . . . . . . . . . . . . .

( 2)avg j−ω( 1)avg j−ω

( )avg jω

1 0 1 2

( )/ 3( ,0) ( ) ( 1)2

hallavg t avg

T jj j a jt tπ

ω ω ω−

−

⎧ ⎫⎪ ⎪⎪ ⎪= = = + −⎨ ⎬⎪ ⎪−⎪ ⎪⎩ ⎭12

( 1, )t j kω ω−

= −

Midpoint speed

1t− 12

t−

0t

( , ) ( ,0) ( 1)avg sj k j a j Tω ω= + −

Estimated positionActual position

( , )est j kθ

( )real jθ

( 1)real jθ −

60

( )hallT j

1t

[ ]1,0( 1, ) ( 1)est real

jj k jθ θ

−

− = −

[ ]1,1( 1,1)est

jjθ

−

−

[ ]1, 1( 1, 1)est

j kj kθ

− −

− −

[ ], 0j[ ],

( , )est

j kj kθ

( , ) ( )est realj k jθ θ=

[ ]1,0( 1,0)est

jjθ

+

+

. . .

Hall-Effect position

signals at every60 electrical

degrees

Average estimated

speed

Actual speed

Estimated speed

Fig. 6.4. Speed and position estimation using low-resolution position sensor [54].

132

1 2

1 2 / 2

t

mt P

ωω

−

−

= in mechanical rad/sec (6.4)

where is the pole numbers of the motor. P

At each 60 electrical degree interval time the speed of the rotor including the

average acceleration or deceleration rate can be defined as:

1 2

( )( ,0) ( 1)2

hallest t avg

T jj a jω ω−

= + − (6.5)

To resemble the high-resolution position sensing phenomenon, the rotor position

and speed corresponding to the well-known Newton’s motion law between two

consecutive hall-effect signals can be estimated as follows:

( , ) ( ,0) ( 1)est est avg sj k j a j Tω ω= + − (6.6)

where sT is the sampling time of the overall system.

At every 60 electrical degrees (hall-effect sensor signal observed) the average

rotor acceleration or deceleration rate (… , , …) is calculated as: ( 1)avga j− ( )avga j

1

( ,0) (( 1),0)( ) est estavg

j j

j ja jt t

ω ω

−

− −=

− electrical rad/s2 (6.7)

where is the time at (( and is the time at 1jt − 1),0)j− jt ( ,0)j as shown in Fig. 6.4. The

present estimated position between two consecutive hall-effect sensor signals, ( , )est j kθ ,

when the acceleration or deceleration is ignored can be expressed as:

( , ) ( ) ( , )est real Pj k jθ θ θ= + j k

s

(6.8)

where denotes the incremental position added to the real position

which is obtained from the hall-effect sensor. Replacing that into (6.8), we obtain:

( , ) ( )P avgj k j kTθ ω=

( , ) ( ) ( )est real avg sj k j j kTθ θ ω= + (6.9)

133

Using angular velocity, ( , )est j kω , the present estimated position, ( , )est j kθ , is

calculated by adding the real position, ( )real jθ , obtained from the hall-effect sensors to

the estimated incremental position, , and also it is the angle rotated from the actual

position to the present that is represented by:

( )P kθ

( )21( , ) ( ) ( 1) ( , )2est real avg s est sj k j a j kT j k kTθ θ ω= + − + (6.10)

The real position, ( )real jθ , can be expressed when as: 0k =

( ) ( ,0)real estj jθ θ= (6.11)

where in the above equations is the index pointer representing the incremental value

of the sampling time,

k

sT , starting at 0 incrementing by one.

Basically, in the above method position is collected from the hall-effect sensors

and the average acceleration or deceleration rate of the rotor is calculated at every 60

electrical degrees interval. The estimated rotor position and the speed values between

those intervals are obtained by using Newton’s motion law equations given above.

Fig. 6.5 depicts the acceleration and speed estimations in each sector and at the sector

boundaries of the proposed DTC scheme, respectively. Because the acceleration or

deceleration of the motor is included in the above position and speed estimation

equations, the hesitation problems associated with step and ramp speed references under

initial load condition are sufficiently improved.

Even though the rotor position is calculated from (6.10), it needs to be clarified

that if the estimated rotor position is still within the assumed section. For example, if the

rotor is assumed to be in the Sector 1, the condition that should be

satisfied. The estimated rotor position is corrected when the limits exceeded for Sector 1.

The correction logic that is used in the Sector 1 is given below [55]:

330 30rθ≤ ≤

If , then ; 330rθ < 330rθ =

If , then . 30rθ > 30rθ =

There are four major advantages associated with the proposed algorithm [55]:

134

• No parameter sensitivity problem because no machine or drive

parameters are used in the rotor position estimation.

• No error accumulation problem as the rotor position is reset at every 60

electrical degrees.

• No starting problem because the hall-effect sensors provides enough

information for correct starting. There will be no wrong direction start.

• Low cost contrary to expensive high precision position sensors such as

optical encoders and resolvers and high reliability as compared to

encoders and resolvers.

• Simple algorithm can be easily implemented by a microprocessor or DSP.

As the rotor position corrects itself at every section transition, the position

estimation error is very reasonable at high speeds and even during dynamic transients.

The estimation algorithm proposed in this thesis guarantees the estimated

position with high accuracy at low speeds.

D

Q

30

aHbH

cH

bH ′aH ′

cH ′

Hall-1Hall-2

Hall-3

1a

2a3a

4a

5a 6a

1 1, tω

2 2, tω

3 3, tω

4 4, tω

5 5, tω

6 6, tω

7 7, tω

Fig. 6.5. Representation of acceleration and speed at the sectors and at the hall-effect

sensors, respectively.

135

CHAPTER VII

AGITATION PERIOD SPEED CONTROL OF VERTICAL-AXIS

LAUNDRY MACHINE

7.1. Modern Washing Machines: Top Loaded (Vertical-axis) vs. Front Loaded

(Horizontal-axis) Washing Machine

Contemporary washing machines are available in two main configurations:

“top-loading” and “front-loading”. The “top-loading” design, the most popular in the

US, Australia, and some parts of Europe, has the clothes placed in a vertically-mounted

cylinder with a propeller-like agitator in the center of the bottom of the cylinder. The

cloths are loaded at the top of the machine, which is covered with a hinged door. The

“front-loading” design, the most popular in Europe (especially in UK), instead mounts

the cylinder horizontally, that allows loading through a glass door at the front of

machine. The cylinder is called the drum.

Tests comparing front-loading and top-loading machines have shown that, in

general, front-loaders wash cloths more thoroughly, cause less wear, and use less water

and energy than top-loaders. As a result of using less water, they require less detergent.

They also allow a dryer to be mounted directly above the washer, which is impossible

with a top-loader. On the other hand, top loaders complete washing much faster, tend to

cost less for the same capacity machine, and allow clothes to be removed at intermediate

stages of the cycle (for instance, if some clothes within a wash are not to be spun).

136

7.2. Introduction to Speed Control of a BLDC Spindle Motor (48 Pole/36 slot) in

The Agitation Cycle of a Washing Machine

In a vertical-axis washing machine, agitation is supplied by the back-and-forth

rotation of the cylinder (drum) and by gravity. The items are lifted up by paddles in the

drum, and they are then dropped down to the bottom of the drum. This motion forces the

water and detergent solution through the fabric. Although more infrequent, there is also a

variant of the horizontal axis design that is loaded from the top, through a flop in the

circumference of the drum. These machines usually have a shorter cylinder and are

therefore smaller [56].

In the agitation part of a laundry washing cycle, the controller controls the power

supplied to the motor using Pulse Width Modulation (PWM) in order to maintain motor

speed in accordance with a predetermined speed profile. The agitation speed profile can

be categorized into three sections [56]:

(i) Ramp up from standstill to a desired plateau speed (steady-state speed), the

so called “acceleration” or “ramp” region.

(ii) Maintaining the plateau speed for a predetermined period of time. This

section is called the “plateau” region.

(iii) Removing the motor power by turning off the lower switches of three-phase

inverter and coasting the motor towards zero speed. The end of plateau

section is the start of the coast section. Once the speed of the motor drops to

the predetermined reversal speed, which is usually defined as half of the

reference plateau speed, motor will be reversed. If it is necessary, after

reaching the reversal speed, the motor will be turned in reverse direction

when the motor speed drops to zero. Consequently, the time between the end

of plateau section and reversal speed is called the “coast” time and the time

between the reversal speed and the zero speed in one stroke is called the

“pause” time. Altogether the coast and pause time establish the “coast/pause”

region of a stroke.

137

The combination of (i), (ii), and (iii) is the stroke time of the washing machine in

the agitation cycle. At the end of each stroke, the motor direction is reversed. In each

agitation stroke, the stroke time is controlled by adjusting the “coast” (pause time is

constant) phase of each stroke in order to maintain consistent washing performance.

Before starting the agitation washing cycle, all of the above phases (times) determined

by the controller depending on the load condition are chosen using the information given

on the washing machine panel by the customer. Only the ramp, plateau phases and pause

times stay constant in every stroke. The “coast” phase is initiated, however, (after a

predetermined period elapsed in the “plateau” phase) by starting a timer and allowing the

motor to coast until the motor velocity (speed) drops to a predetermined reversal speed

at which point the motor is reversed to begin the next agitation stroke. Several

experiments have proven that the reversal speed should be chosen to be equal to the half

of the plateau speed and can be adjusted depending on the load conditions (bunching,

etc.). Reversal speed will be changed in accordance with the difference between the

timer value when the velocity reaches the reversal speed and the predetermined timer

value representing the total coast time. If the timer value is greater than the desired value

or range of values then the predetermined value of the reversal speed will be increased.

Alternatively, if the timer value is less than the desired value or range of value the

reversal speed will be decreased. Eventually, the period of the coast phase of the next

stroke should be closer to the desired period. The “coast/pause” phase is not going to be

the main focus in this thesis work, instead the “ramp” and “plateau” phases will be the

main subjects to be discussed. A simple flowchart of the agitation speed control is

depicted in Fig. 7.1.

138

Initialize( Set Plateau Speed, Ramp Time, Plateau

Time, Pause Time)

Ramping

Steady-state (Plateau region)

Coasting/Pause

Reversing

Fig. 7.1. Simple flowchart of the commercial agitation speed control [57].

7.3. Agitate Washer Controller Model in SIMPLORER®

A complete agitation washer controller is built in SIMPLORER®. Unlike a

real-time microcontroller that uses interrupts and counters, the simulation will use time

based logic for simplicity. Details are as follows:

As explained before, the agitator speed controller has ramp, plateau, coast and/or

pause time sections in the reference speed profile which is shown in Fig. 7.2. Basically,

the controller has four inputs. Those are: Plateau Speed (PS), Ramp Time (RT), Plateau

Time (PT), and Pause Time (PaT). Coast Time (CT) is an input usually maintained at a

139

certain predefined time by the controller. In this thesis, the coast and pause sections are

not modeled fully. They were discussed briefly in the previous section. Characteristics of

each section will be explained. One positive complete cycle (stroke) is shown in the

graph below. The negative cycle which has the same shape, but in reverse direction, is

not shown here, the reverse rotation starts when the cost/pause section is over.

t(ms)

Plateau Speed

Ramp Time Plateau Time

Pause Time/Coast Timewref* (rpm)

1 2 3

Fig. 7.2. Reference speed profile for the agitation speed control in a washing machine.

The operation of the agitation control in a laundry washing machine algorithm

used in commercial washing machines is discussed below. A flowchart is provided

detailing the operation of the simple agitation washing controller which is implemented

in the simulation platform, SIMPLORER®.

140

Accelerate motor to plateau speed

START

Overshoot compensation,attain and maintain plateau speed

Plateau periodcompleted?

Remove motor powerStart timer. TIME=0

Motor speed = reversalspeed?

Stop timerTimer = elapsed coast time

Elapsed coast time > preferred coast

period (range)?

Elapsed coast time < preferred coast

period (range)?

Increment reversal speed Decrement reversal speed

Agitation cyclecompleted

Reversal motor direction

Stop agitating

Fig.7.3. Detailed flowchart of the simple agitation speed control of a washing machine

[58].

141

Fig. 7.3 presents the instructions required to carry out the agitation phase of a

washing cycle with a detail flowchart. Descriptions of that flowchart are as follows:

When the washing machine has completed a fib phase and the laundry load is

floating (or submerged) in a suitable mixture of water and detergent, the motor is

accelerated (ramped up) incrementally by increasing the PWM duty cycle starting at a

minimum value up to plateau speed. More details about ramping up the motor to the

desired plateau speed will be discussed later.

Due to the inertia of the drum and laundry load speed overshoot can occur,

therefore an overshoot compensation algorithm is developed to bring the exceeded motor

speed down to the desired plateau speed as quickly as possible. When the plateau speed

is achieved after applying the overshoot compensation (if it is necessary) then the

desired plateau speed is maintained till the predetermined “plateau” region is completed

(by checking the motor controller timer). At the end of the plateau region the motor

power is removed by turning off the lower side of the inverter switches. At the same

time the motor timer is started from zero to determine the length of the coast section.

While the coasting condition is continuing the reversal speed may be reached (which is

assumed to be half of the plateau speed). As time elapses during this phase, the total

coast region length is determined for use in the adaptation mechanism of the next

strokes. The elapsed coast time is then compared with the predetermined coast time. If

the measured coast time is greater than the desired coast period (or range of periods)

then the controller increments the reversal speed to be used in the next stroke. If the

measured coast time is less than the desired coast period the controller decrements the

reversal speed for the next stroke. One stroke late adaptation algorithm is therefore

established by controlling the coast time because if the reversal speed for the next cycle

is higher than what was defined previously then the coast time will be shorter and vice

versa, eventually causing a change in the length of the stroke time. If the pause time is

defined as zero then the agitation reverses the motor and the same method is applied

until the agitation phase of the washing cycle is completed. Agitation is then stopped and

the control executes the next washing cycle. If, however the agitation phase is not yet

142

complete, then the motor direction is reversed and power is reapplied to the motor

windings where the above described process is repeated until the end of the agitation

phase [56].

1) Ramp-up Section:

As it is known in DC motor control, to increase the rotational speed of the motor

the PWM duty cycle of the inverter outputs should be increased. To reduce the speed of

the motor the duty cycle should be decreased, as a result it can be stated that the change

in the duty cycle is proportional to the change in speed. The duty cycle of the PWM

signal which is applied to the motor terminals by a VSI inverter is determined by the

following equation:

max

_ (%) 100 / currentPWMduty cyclePWM

⎛ ⎞⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜⎝ ⎠ (7.1)

The ratio above determines the on time of the one PWM period (PWM frequency

can be any value not less than 2 kHz). If becomes equal to the full

DC-link voltage will be applied to the motor terminals and maximum speed will be

achieved. In starting, is chosen as small as possible to handle the light load.

is the key component to decide the resolution of the speed increment. If it is

higher and the incremental rate of is small then a higher speed increment

resolution is achieved, but if it is small and the incremental rate is bigger

then a lower speed increment resolution is provided.

currentPWM maxPWM

currentPWM

maxPWM

currentPWM

currentPWM

In ramping, the PWM rate is incremented till the end of ramp period. At that

moment the speed is measured by the help of three hall-effect sensors which provide

electrical signals indicative of the rotor position. These signals are then fed to the input

of a microcontroller via an interface circuit. According to the resultant speed feedback,

(along with the duty cycle) is adjusted up or down depending on whether the

measured speed is too low or too high.

currentPWM

143

As several tests show, regardless of how fast the agitation period is, it is possible

to have any ramp period from (e.g. 80 ms – 400 ms) for a better speed profile with less

overshoot under heavy laundry load conditions. The ramp time also depends on the wash

profile; if it is very gentle then soft agitation is desired therefore a longer ramp up time is

preferred and vice versa.

plateauV

PWM increments

currentPWM

PT

( )Time s

( )Speed RPM

Fig. 7.4. PWM increment during the ramp time [58].

currentPWM is incremented at every PT (sampling time of the in

ramping) as shown in Fig. 7.4 in a series of incremental steps, therefore the speed of the

motor gradually increases because the PWM increment occurs at every time interval

currentPWM

PT

as shown in Fig. 7.5.

144

plateauV

currentPWM

PT

( )Time s

( )Speed RPM

RampT

Fig. 7.5. Speed response during ramp up when PWM is incremented at every PT time

[58].

In Fig. 7.4 the angular speed is shown diagrammatically as increasing in discrete

steps. Each step corresponds to an increase in which also increases the duty

cycle of the PWM signal. The smaller

currentPWM

PT is the more is incremented

therefore more power is supplied to the motor. All the above descriptions given about

ramping up are summarized as follows: is started at a certain number which

can handle the light laundry load at starting and a certain increment step value is chosen

(e.g. 16) such that the step time,

currentPWM

PWMcurrent

PT , (also known as the sampling time) of

can be found if the ramp time is known by the following formula:

currentPWM

_P

Ramp TimeTSTEPS

= (7.2)

At every sampling time the value is incremented by a certain

predetermined amount until the speed reaches the desired plateau speed in the

predetermined ramp time. If the actual speed is less than what is necessary then ramp

currentPWM

145

adjustment is used starting from the very next stroke. The ramp adjustment algorithm is

a mandatory tool in agitation control. For example, the load on the motor can not always

be the same due to varying wash loads, and it is not possible to predetermine the number

of incremental steps in order to obtain the correct speed within the ramp time.

A small amount of power (i.e. a small duty cycle value) is enough to start

accelerating the motor to the desired plateau speed during the ramp time when there is a

very light load on the machine. With a heavy load more power (larger duty cycle value)

will be required. The amount of power that can be supplied to accelerate the motor to the

desired plateau speed in the predetermined ramp time is proportional to the rate of

increase of the incremental steps. The number of steps is initially preset to a minimum

setting so that no adjustment is needed for very light loads. Let’s say a machine has a

very light load and 6 incremental steps are necessary to accelerate the motor to a desired

plateau speed in a predetermined ramp time (e.g. 100 ms). On the other hand with a

heavy load 62 steps might be required to reach the same speed in same amount of time

(100 ms). The software therefore needs to adjust the number of steps to attain the desired

speed at a desired interval time. This is performed when the speed is first time monitored

at the end of the ramp time at every stroke. If the measured speed at the end of the ramp

period is found to be under the desired plateau speed then the number of steps is

incremented, thus the value of increases, and hence the torque of the motor

increases [58].

currentPWM

2) Plateau Section:

Ideally, in the plateau region, a constant speed is maintained for a desired

predetermined plateau time period. This is the case when there is no heavy load (too

much laundry load) on the motor, but at extremely heavy loads a normal agitate

controller cannot maintain the speed at the desired plateau speed due to torque

constraints on the motor. There is a need for overshoot control in the plateau region to

ensure the speed maintains an ideal profile at heavy loads, as shown in Fig. 7.6.

146

plateauV

( )Time s

12 plateauT 1

2 plateauT

plateauT

( )Speed RPM

Fig. 7.6. Ideal speed profile of the agitation speed control [58].

Under very heavy loads the actual profile digresses from the desired speed

profile as shown in Fig. 7.7. Several experiments show that the load condition of the

motor during the plateau region is not its heaviest until approximately half way through

the plateau region. If the load in the drum is increased, then the tail end of the plateau

region begins to diverge ¾ of the way through the plateau region, hence the best point to

measure the speed and determine when to add or subtract compensation is at the mid

point of the plateau region as illustrated in Fig. 7.7 [58].

147

plateauV

( )Time s

plateauT

tΔ

plateauVΔ

( )Speed RPM

Fig. 7.7. Typical practical speed response of the agitation speed control [58].

If the measuring point indicates a speed that is under the desired plateau speed

then on the next stroke cycle at the end of the ramp period the software will increase the

plateau speed by an error received from the compensation algorithm, Δ , for a

predetermined time, , as seen in Fig. 7.7. As the load increases, the compensation

should also increase. The first parameter that is increased is the overshoot time, .

This is increased until either the speed has reached a desired value at the measuring point

(mid point of the plateau region) or Δ is equal to the plateau time [58].

plateauV

tΔ

tΔ

t

If is equal to the plateau time and the speed at the measuring mid point is still

under the desired speed due to very heavy load, then the overshoot speed, , is

increased until a predetermined limit is reached or the speed at the measuring mid point

is acquired [58].

tΔ

plateauVΔ

If the maximum compensation limit has occurred but actual speed measured at

the mid point of the plateau region is still under the desired plateau speed due to heavy

load condition, however, then the ramp rate is increased by increasing the number of

steps, as explained above, in relation to the acceleration ramp control.

148

CHAPTER VIII

EXPERIMENTAL AND SIMULATION ANALYSIS

8.1. Open-Loop Simulation vs. Experimental Results

In this section of Chapter VIII, before going into the detailed discussion of the

simulation and experimental results of the commercialized agitation cycle speed control,

an open-loop steady-state and transient-state simulation and experimental results will be

given and those results discussed and compared. The experimental setup of the

open-loop control system is explained in detail. Important motor parameters such as

efficiency and power factor, are measured in both the simulation and experiment.

Open-loop analysis and suggested measurement methods of those motor parameters are

explained in detail.

8.1.1. Open-Loop Steady-State and Transient Analysis (Simulation)

Open-loop (V / f control) MATLAB/SIMULINK® simulations have been

conducted to verify and compare the performance of the simulated 48 pole PMSM with

the real model. All the open-loop steady-state and transient state simulations are carried

out using the default variable-step ode45 (Dormand-Prince) algorithm. The maximum

step size, relative tolerance and absolute tolerance are all set to 1e-3. Several parameters

such as power factor, efficiency, rms motor phase current, and load angle are simulated

at a 100 rpm (40 Hz) steady-state operating speed to compare with the experimental

results to verify how close the simulated motor model is to the real motor. A three-phase

AC voltage source is used to implement the open-loop V / f control in the simulation of

the PMSM.

149

The PMSM cannot be started at reasonable high constant frequencies, such as

40 Hz, which is used in our tests, so in the simulations the initial speed is selected as

40 Hz (100 rpm) instead of 0 Hz to overcome this problem. The V / f ratio is set to 2,

meaning an 80 Vrms sinewave signal with 40 Hz frequency will be generated. By

choosing V / f ~= 2, a good speed result (no pulsation) and better current shapes (close

to pure sinusoidal and less oscillation) are obtained, and the results of efficiency, power

factor, rms motor phase current (Phase A) and motor speed versus load torque are

displayed in Fig. 8.1 in comparison with the experiments. It is observed that the

simulation results are very close to the experiments, especially at low load torque values

since the parameter change effect is not considered in the simulations when the load

torque increases up to 10 N·m a discrepancy occurs at high temperature (rotor flux

linkage amplitude and resistance are affected by temperature change). If the saturation

level of the motor is reached and exceeded then the synchronous inductance value will

also alter.

A transient analysis is performed to observe the starting performance of the

simulated model in comparison with the experimental results; basically this test has been

done to verify the inertia and other torque effective components such as friction and

windage load etc. on the transient conditions. From Figs. 8.2 and 8.3, it can be seen that

the simulation speed response to the step input is faster than the experimental result.

Having a transient response in open-loop is quite challenging due to the lack of position

information to synchronize the rotor field with stator rotating field at start up. If the

transient is less than 0.4 second both responses converge and reach the steady-state

speed. To start the motor properly in both simulation and experimental transient tests,

the zero initial position is selected as the reference point. Any position can be chosen as

a reference, but since the input voltage is a sinewave, starting the motor from the zero

position generates less torque pulsation. In general, the zero reference position is the

position when the rotor flux is aligned with the Phase A axis. In practice, if the neutral

access is available then enough DC voltage is applied between Phase A and the neutral

point to enable the rotor to stay aligned with the Phase A axis. If the neutral is not

150

available then Phase B and C are short-circuited, so when DC voltage is applied between

Phase A and the short-circuited Phase B and C the same zero position alignment will be

achieved.

100 RPM (40 Hz) Open-Loop (V/f=2) Experimental vs. Simulation Results (80Vrms Sinewave Input,Ld=Lq=92.7274mH)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 2 4 6 8 10 12

Torque (Nm)

Eff,

PF,Ia

0

30

60

90

120

150

Speed

Exp. Efficiency Exp. Ia (Amp) Exp. PF Sim Ia (rms) Sim. Eff Sim. PF Exp. Speed (RPM)

Fig. 8.1. Efficiency, power factor, motor current (Phase A), and motor speed versus

applied load torque (open-loop exp. and sim.) at 100 rpm with V / ~= 2. f

151

0-37.5 RPM Experimental and Simulation Dynamic Speed Responds (V/f=18/15)

0

5

10

15

20

25

30

35

40

45

50

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time (s)

Spee

d (R

PM)

Exp.Speed Sim.Speed

Fig. 8.2. Dynamic speed response of exp. and sim. when V / f = 18/15 (0-37.5 rpm).

0-37.5 RPM Experimental and Simulation Dynamic Speed Responds (V/f=1)

0

5

10

15

20

25

30

35

40

45

50

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time (s)

Spee

d (R

PM)

Exp.Speed Sim.Speed

Fig. 8.3. Dynamic speed response of exp. and sim. when V / f = 1 (0-37.5 rpm).

152

8.1.2. Open-Loop Steady-State and Transient Analysis (Experiment)

Experimental tests have been conducted on a 325 PM synchronous machine

(sinusoidal back-EMF) with 48 rotor poles. Even though the agitation control algorithm

generates squarewave currents to control the motor, in an open-loop test the sinewave

input voltages are used to get better performance on the measurements of the efficiency,

power factor, load torque, and rms phase current of the motor. Since the current is

rectangular in agitation control, but the back-EMF of the machine is sinusoidal, the

resultant torque (electromagnetic torque) is not smooth. Fig. 8.4 shows the

electromagnetic torque of a PMSM having a sinusoidal back-EMF when it is driven with

square-wave currents.

dcV

Time (s)

Torque (N.m.)

Fig. 8.4. Torque response of PMSM when it is supplied by quasi-square wave currents.

In the open-loop transient test, the rotor is locked with the zero position as

explained above and a three phase programmable AC voltage source is connected to the

motor terminals. The minimum frequency available from the power source, 12 Hz, is

chosen such that the V / f ratio will be around 1. With this V / f ratio the rms value of

the sinusoidal voltage becomes 12 Vrms. Additionally, a second test for transient

analysis is performed with V / f ~= 18/12. During this test, a dynamometer controller

(DSP600) is used so the speed response can be stored in the M-TEST 4.0® software and

saved as a Microsoft® Excel spreadsheet as shown in Figs. 8.2 and 8.3.

The open-loop efficiency calculation, explained in Chapter IV, which is done in

MATLAB/SIMULINK® shows an agreement with the actual measurement provided by

the Magtrol Power Analyzer Model 6530, as shown in Fig. 8.1. Measurements of real

153

power, power factor, efficiency, motor voltages and currents, etc. are obtained in the

M-TEST 4.0® software which communicates with the power analyzer through the GPIB

cable. Other necessary equipment for measuring the above parameters include a Magtrol

Hysteresis Dynamometer (Model HD-800-6N), a Magtrol DSP6000 Dynamometer

Controller interconnected with the Magtrol Hysteresis Dynamometer and the

M-TEST 4.0® software. The software sends the load torque information to the Magtrol

Hysteresis Dynamometer through the Dynamometer Controller and depending on the

control configuration (either closed or open-loop (if curve test is selected in the M-TEST

4.0® software, then the dynamometer controller will control the values assigned in the

software in a close-loop fashion) the load torque value is controlled by the Dynamometer

Controller. Communication between the M-TEST 4.0® software and the dynamometer

controller is achieved through the GPIB cable. The results of the measured and

calculated parameters which are processed in the power analyzer are sent back to the

M-TEST 4.0® software to be read. Fig. 8.5 shows the measured experimental efficiency,

power factor, rms motor phase current and motor speed versus applied load torque when

the curve test is used. More detail about curve test will be given later in this chapter.

154

100 RPM (40Hz) Open-Loop (V/f=2) Experimental Results (80Vrms Sinewave Input)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 2 4 6 8 10 12

Torque (Nm)

Eff,P

F,Ia

0

30

60

90

120

150

Speed

Exp. Efficiency Exp. Ia (Amp) Exp. PF Exp. Speed (RPM)

Fig. 8.5. Efficiency, power factor, motor current (Phase A), and motor speed versus

applied load torque (experimental, open-loop) at 100 rpm with V / f ~= 2.

8.1.3. System Configuration for Electrical Parameters Measurements

A Magtrol Hysteresis Dynamometer provides motor loading with Magtrol

Programmable Dynamometer Controller (DSP6000) acting as the interface between the

personal computer running M-TEST 4.0®. The Hysteresis Dynamometer and motor

electrical parameters are measured or used to determine load points by a Magtrol Power

Analyzer. A three-phase Power Analyzer Data Acquisition system is configured such

that it can obtain data on each individual phase and/or their sum which is used in the

chosen parameters (amps, volts, input watts and power factor).

The interface between the computer and the electronic instrumentation is

provided by the National Instruments™ PCI-GPIB card when using a DSP6000.

Basic block diagram of overall system for efficiency measurements is

represented in Fig. 8.6. Fig. 8.7 illustrates the configuration of the overall measurement

system including one additional power analyzer (magtrol model 6550). There are two

reasons for having an additional power analyzer: The first reason is to measure the input

155

active power of the single phase supply source so that the overall system efficiency can

be easily calculated. The overall efficiency is represented as in the same figure.

Efficiency loss in the power converter can be calculated by using the efficiency of the

three-phase inverter measured by power analyzer model 6530 when the motor is driven

by the agitation washer controller. To achieve that, one of the power analyzers (model

6550) measures the active power which goes to the single phase rectifier from the single

phase supply while the other (model 6530) measures the active power which goes to the

motor provided by three-phase inverter. The output power generated by motor is the

same when those measurements are taken place. A detailed system connection is given

in Fig. 8.7.

1η

Magtrol 6550 Magtrol 6530Motor Controller

+Inverter

HDPMSM

Pin(1) Pin(2) Pout

1(1)

out

in

PP

η = 2(2)

out

in

PP

η =

Fig. 8.6. Basic block diagram of overall system for efficiency measurements.

156

Transformer

MotorController + Inverter

PCM C2Interface

Unit

GPIBCard

GPIBCable

Single PhasePower Source

Variac

GPIBCable

Single PhasePowerPin(1)

GPIBCable

DSP6000Programmable Dynamometer

Controller

Magtrol 6550Power Analyzer

Magtrol 6530Power Analyzer

Three PhaseSource

Pin(2)

Three PhaseSource

PMSMPout

M-TEST4.0 Hall-effect

Circuit

COMPort

Hysteresis Dynamometer

Fig. 8.7. Overall system configuration.

8.1.4. The M-Test 4.0 Software

Magtrol's M-TEST 4.0® software is a state-of-the-art motor testing program for

Windows®-based data acquisition. Used in conjunction with Magtrol's Motor Testing

Equipment, M-TEST 4.0® provides testing and data options to help determine the

performance and characteristics of a motor under test.

The data generated by the program can then be stored, displayed and printed in

tabular or graphic formats, and easily imported into a spreadsheet (Microsoft® Excel).

M-TEST 4.0® is ideal for tasks such as simulating loads, cycling the unit under test and

motor ramping.

In our tests, M-TEST 4.0® is equipped to work in conjunction with the following

Magtrol motor testing instruments:

157

• Dynamometer Controller (DSP6000)

• Hysteresis Dynamometer

• Power Analyzer (Model 6530 and 6550)

8.1.5. Curve Testing

M-TEST 4.0® can be used in a way that simulates complex load profiles. This

may be for a heat run or endurance testing, simulating a real life usage, or just for

checking a few specific data points.

Loading may be accomplished by closing a control loop on Speed, Torque, or

Output Watts, plus Amps or Input Watts if a power analyzer is incorporated into the

system. Because closed loop speed and torque are internal functions of the dynamometer

controller, the control loops are very fast and highly controllable. The remaining

functions use a routine in M-TEST 4.0® to close the loop and provide control. These will

not provide as tight a control as the internal functions, but they are quite satisfactory for

most applications.

Loading can be accomplished by either stepping or ramping to the desired point.

If a load point is wanted to be stepped up suddenly, then a time of “0” (zero) should be

entered for that point. If a ramp to a load point is desired, then the number of seconds

should be entered (or minutes, depending on the timebase setting) for the controller to

ramp from the starting point to the ending point. To remain at a fixed load for a period of

time, the same value for "From" and "To" is used. Any values entered in the “From” and

“To” columns will be in the units specified by the control parameter.

To obtain the desired electrical parameters, such as power factor, efficiency, and

etc. in open-loop control, the following torque curve test is performed depicted as a table

in Fig. 8.8:

158

Sequence From To Time Description1 0 0 10 Dwell at 0 N.m. for 10 seconds2 0 10 10 Ramp the torque from zero to 10 in 10 seconds3 10 10 10 Dwell at 10 N.m. for 10 seconds4 10 0 30 Ramp down to zero in 30 seconds5 0 0 10 Dwell at 0 N.m. for 10 seconds

Fig. 8.8. Sample curve test.

And the graphical representation of the above torque curve test is given in

Fig. 8.9.

10 20 30 60

Time (s)

Torque (N·m)

10

Fig. 8.9. Graphical representation of the sample curve test.

8.2. Experimental vs. Simulation Results (Commercial Agitation Control)

In this section of Chapter VIII, the simulation and experimental results of the

commercialized agitation cycle speed control are given and those results are discussed

and compared. Several important motor parameters, such as efficiency and power factor

as well as line current, motor phase currents, DC-link voltage and motor speeds, are

obtained both in simulation and in experimentation and they are compared.

159

8.2.1. Complete Agitation Speed Control Analysis (Steady-State)

First, to compare with the experimental steady-state results, a steady state

simulation model is built in SIMPLORER®. Validation is achieved by comparing several

parameters such as line current observed in the single phase rectifier as depicted in

Fig. 8.10 , overall efficiency (input real power is measured in the experiment and

calculated in the simulation from a single phase supply source, and power factor. The

input line current obtained from the simulation under 10 N·m of load torque at steady

state (100 rpm) closely resembles the real-time line current. This concludes one section

of the validation process between the SIMPLORER® agitation steady-state model and

the real-time agitation system.

Exp vs. Sim. Line Currents (Is) @ 10Nm

-8

-6

-4

-2

0

2

4

6

8

0,31 0,315 0,32 0,325 0,33 0,335 0,34 0,345 0,35

Time (s)

Cur

rent

(A)

Sim.Is(A) Exp.Is(A)

Fig. 8.10. Input current of the single phase diode rectifier at 10 N·m of load torque.

Fig. 8.10 represents the input currents (simulation and experiment) of the single

phase diode rectifier fed by input voltage source from the wall plug. It is captured using

a LeCroy Digital Oscilloscope at 10 N·m of load torque when the motor runs at 100 rpm.

The experimental input currents in Figs. 8.10 and 8.11 are identical, but the currents

160

shown in Fig. 8.10 are exported from SIMPLORER® and from the LeCroy DSO to

Microsoft® Excel for a better comparison of the simulated and experimental input

currents.

Fig. 8.11. Input current to the single phase diode rectifier (experiment) at 10 N·m of load

torque (100 rpm).

The voltage source feeding the single phase diode rectifier is obtained from the

wall plug which is measured as 122.1 Vrms. This value is used in the simulations as

shown in Fig. 8.12.

161

Source Voltage (V)

-200

-150

-100

-50

0

50

100

150

200

0 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09 0,1

Time (s)

Volta

ge (V

)

Fig. 8.12. Voltage source obtained from the wall plug.

Fig. 8.13 shows the DC-link voltage of the three-phase inverter when the

122.1 Vrms sinusoidal input voltage is applied to the single phase diode rectifier.

162

Dc-Link Voltage (V) @ 10 N.m Load Torque (100 RPM)

310

315

320

325

330

335

340

0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,5

Time (s)

Volta

ge (V

)

Fig. 8.13. DC-link voltage of the VSI at 10 N·m of load torque (100 rpm).

Fig. 8.14. indicates the simulated steady-state speed and experimental speed at

10 N·m of load torque. It can be seen that the speed responses are very similar. Due to

the large sampling time provided by the dynamometer controller, (0.1 second in the

curve test; more detail about curve testing will be discussed later in this chapter) the

real-time speed fluctuates more than the simulated one. Additionally, the dynamometer

has a 60-bit encoder inside which only has a resolution of 0.5 rpm for the speed

calculation. For example, if the actual speed is 100.3 rpm, the dynamometer velocity

sensor rounds it up to 100.5 rpm, therefore the real-time speed observed is almost

constant all the time except when there is a step change in the speed that is greater than

0.5 rpm. On the other hand the simulation speed result has a higher resolution so it

shows more speed ripples.

163

Load Torque and Plateau Speed (100 RPM) vs Time (s) @ 10 N.m Load Torque (Steady-State)

-2

0

2

4

6

8

10

12

90,1 90,3 90,5 90,7 90,9 91,1 91,3

Time (s)

Torq

ue (N

.m)

90

92

94

96

98

100

102

104

106

Speed (RPM

)

Load Torque (Nm) Exp.Speed (RPM) Sim.Speed(RPM)

Fig. 8.14. Steady-state load torque and plateau speeds (experiment and simulation) at

10 N·m of load torque (100 rpm).

The motor is assumed to be rotating at a constant speed, which is chosen as

100 rpm in our test, therefore in steady-state simulation, the transient is ignored by

setting the initial motor speed to 100 rpm in the motor parameters part of the

SIMPLORER® built-in PMSM model. The reason that 100 rpm is chosen is because it is

the average agitation control steady-state speed used in many commercial top-loaded

washing machine drives; therefore in all simulations 100 rpm is chosen as a reference

speed value.

164

Steady-State Electromagnetic Torque @ 10 N.m Load Torque (Simulation, 100 RPM)

-2

0

2

4

6

8

10

12

14

0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,5

Time (s)

Torq

ue (N

.m)

Fig. 8.15. Steady-state electromagnetic torque (simulation) at 10 N·m of load torque

(100 rpm).

In steady-state, the electromagnetic torque produced by the motor is obtained as

shown in Fig. 8.15. If other effective torque components are ignored in the

electromagnetic torque equation of the motor, such as viscous damping coefficients and

etc., then the average torque is almost equal to the load torque, which is 10 N·m in this

case. When the time scale of Fig. 8.15 is increased, the resultant torque has excessive

pulsation mainly due to the shape of the excitation currents and the back-EMF waveform

of the motor. Since this motor has a sinusoidal back-EMF, if the excitation is provided

with square wave currents then the multiplication of these signals, which is the resultant

electromagnetic torque, will generate more torque pulsation than that of the sinusoidal

current excitation. If less torque pulsation is desired then it is better to energize this

motor with sinusoidal currents. That is why sinusoidal excitation under the DTC scheme

165

is proposed in Chapter VI. The results of the proposed method are given in Section 8.2.2

of this chapter and those results are discussed.

Overall efficiency is obtained in a way that input power is calculated (in

simulation) from the input single phase supply source and the output power is calculated

(in simulation) from the known electromagnetic torque of the motor as illustrated in

Fig. 8.16. Obtaining the overall efficiency for real-time is performed by using several

pieces of equipment such as the Magtrol power analyzer model 6530, the dynamometer

controller model DSP 6000, and the hysteresis dynamometer model HD-800-6N.

Speed Controller

PWM Voltage Signal

Generator VSI

Vdc

PMBLDCM~3

Rotor Position

(Resolution of 60° electrical degrees)

Dwref*

t(s)

Plateau Speed

Ramp Time Plateau Time

Pause Time/Coast Timewref* (rpm)

va*vb*vc*

+ -

Pout

Pin PinPout

=η

L:3 H:85

100 rpm steady state pull up and down test is done for overall system eficiency verification

rθrθ

rθ

Fig. 8.16. Block diagram of the overall commercialized agitation washer control with the

trapezoidal voltage waveform and efficiency calculation and measurement.

Experimental and simulated three phase excitation currents are shown in

Figs. 8.17 and 8.18, respectively. The simulated three phase square wave motor currents

resemble the experimental currents and their amplitudes are quite similar to each other.

This validates the preciseness of the simulated model when compared with the real

control system.

166

Fig. 8.17. Experimental three-phase motor phase currents at 10 N·m of load torque

(100 rpm).

Fig 8.19 shows a comparison of the simulation and experimental results of the

efficiency and power factor in a commercialized overall agitation speed controller. As

expected, the power factor and efficiency of the simulation are greater than the

experimental results.

167

Three Phase Motor Currents @ 10 N.m Load Torque (100 RPM)

-2

-1,5

-1

-0,5

0

0,5

1

1,5

2

0,2 0,21 0,22 0,23 0,24 0,25 0,26 0,27 0,28 0,29 0,3

Time (s)

Cur

rent

(A)

Phase A (A) Phase B (A) Phase C (A)

Fig. 8.18. Simulated three-phase motor currents at 10 N·m of load torque (100 rpm).

Exp. PF. , Eff. and Sim. PF., Eff. vs. Torque (N.m) @10Nm (100RPM steady state)

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0 2 4 6 8 10 12

Torque (Nm)

Eff.,

PF.

80

85

90

95

100

105

110

115

120

Speed (RPM

)

Exp.PF.(x100%) Exp.Eff. (x100%) Sim.PF. (x100%) Sim.Eff. (x100%) Exp. Speed (RPM)

Fig. 8.19. Exp. PF., eff., and sim. PF., eff. vs. torque at 10 N·m (100 rpm).

168

Second, to compare with the experimental transient results, a transient model

(ramp up) is built again in SIMPLORER® and added to the steady state model such that

complete agitation model is achieved. A simplified block diagram of the overall

commercialized agitation washer control is displayed in Fig. 8.20. The speed controller

block, shown in Fig. 8.20, is not the typical PI or PID type compensator. It acts as a P

controller or what you might call a hysteresis controller. Speed regulation is achieved by

the help of three hall-effect sensors. The time difference between two consecutive

hall-effect signals is the key of speed sensing in this simple agitation speed controller.

Depending on the difference in the measured speed (in some sense the estimated speed)

and the reference speed, the PWM duty cycle value is adjusted (either increased or

decreased) accordingly, but it is kept lower than the maximum allowable PWM value,

, by using saturation block as shown in Fig. 8.20. This scenario would occur

mainly during the steady-state region. In the event of ramping, in this case from zero

speed till steady-state speed, (plateau speed) the PWM duty cycle value (starting from a

certain value such that under light load the motor can start up easily) is constantly

increased until ramp time is over. At that moment the motor speed is measured for the

first time and it will be continuously checked as long as the plateau region is active. The

speed is checked through the end of the stroke time until the beginning of the next

stroke.

maxPWM

wref* Speed Controller

PWM Voltage Signal

GeneratorVSI

Vdc

PMBLDCM~3

Rotor Position (Resolution of 60°

electrical degrees)

D

va*

vb*

vc*

+ -

rθrθ

rθ

Fig. 8.20. Block diagram of the overall commercialized agitation washer control with

trapezoidal voltage waveform.

169

8.2.2. Complete Agitation Speed Control Analysis (Transient and Steady-States)

Calculations of some electrical parameters of the control system, such as power

factor and overall efficiency, are done by using the steady-state model. To implement the

agitation algorithm in SIMPLORER®, a ‘state graph’ method is considered to be very

suitable for the simple logical based agitation algorithm. State graphs are used in

conjunction with transition state blocks which determine whether the state should active

or inactive depending on the logic of the transition blocks. Efficiency and power factor

calculations are performed in two different ways and same results are obtained. First,

SIMPLORER®’s own rms and mean value measurement blocks are used to calculate the

efficiency and power factor in a similar manner to the open-loop

MATLAB/SIMULINK® model. The second way of obtaining those parameters is by

using SIMPLORER®’s DAY Post Processor. After the simulation is run and finished, in

the DAY Post Processor’s main menu, when the <<Power>> option is selected in

SIMULATION>Analysis then the active power, apparent power, reactive power, power

factor, Total Harmonic Distortion (THD), and FFT of the selected currents or voltages

can be gathered. By knowing the active power of the input signal and output power of

the motor, efficiency can be easily calculated.

170

Ramp+Plateau Speed (100RPM) vs time @0Nm

-20

0

20

40

60

80

100

120

0 0.5 1 1.5 2 2.5 3 3.5 4

Exp.Speed (RPM) Sim.Speed(RPM)

Ramp Time=0.6125sec

RPM

Spee

d (r

pm)

ti )me (sTime (s)

Fig. 8.21. Experimental and simulation speed response (transient and steady-state) at

no-load.

The transient and steady state performance are illustrated in Figs. 8.21 through

24. Those figures display the simulation and experimental ramp up and steady state

conditions at the 100 rpm plateau speed under no-load with a 612.5 ms ramp time, under

no-load with a 700 ms ramp time and at under 5 N·m of load torque with a 200 ms ramp

time, respectively. The simulation and experimental results are plotted in one xy-axis to

compare and validate the results easily. The results which were obtained from the

simulation closely resemble the speed response of the experimental tests. Since the speed

control is sluggish, the results are not satisfied. The transient speed response of the

simulations at no-load generate a slightly larger overshoot and settling time as compared

with the experimental ones. The reason for this could be the neglect of some side effects

in the simulation model, such as windage torque, friction on couplings, the

dynamometer’s inertia, etc. These problems were not observed when the motor was

initially loaded and ran at 5 N·m, as shown in Fig. 8.22. The main reason of not

171

observing the oscillation effect on the speed result when 5 N·m load torque was applied

is that 5 N·m of load torque is much larger than the other effective torques, such as

windage torque and friction. On every occasion once the transient state passes, the

steady state results are observed to be much closer to the experiments.

Ramp+Plateau Speed (100RPM) vs time (s) @5Nm

-20

0

20

40

60

80

100

120

0 0.5 1 1.5 2 2.5 3 3.5

Exp.Speed (RPM) Sim.Speed (RPM)

Ramp Time=~0.2sec

RPM

Spee

d (r

pm)

time (s)Time (s)


5 N·m of load torque.

172

Ramp+Plateau Speed (100RPM) vs time @0Nm

-20

0

20

40

60

80

100

120

0 0.5 1 1.5 2 2.5 3 3.5

Sim.Speed(rpm)

4

me (s)

RPM

Exp.Speed (RPM)

Ramp Time=~0.7sec

Spee

d (r

pm)

tiTime (s)


no-load (0.7 second ramp time).

Fig. 8.24 shows the simulation of the rotor position from the beginning of the

transient state when the ramp time is chosen as 660 ms and the plateau speed as 100 rpm

under no-load. It can be seen that when the time and rotor speed increases the sawtooth

periods become more narrow until the steady-state is reached at 0.66 second shown in

Fig. 8.24. Once the steady-state speed is achieved, as seen in Figs. 8.21 through 8.23,

then the sawtooth periods become more identical which means that the speed of the rotor

is constant.

173

Rotor Position (0-2pi)

-1

0

1

2

3

4

5

6

7

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8

Time (s)

Posi

tion

(rad

)

Fig. 8.24. Rotor position obtained in simulation.

Figs. 8.25 and 8.26 indicate the steady-state and the combined transient - steady-

state SIMPLORER® models of the commercialized agitation washer controller,

respectively.

174

D1

D2

D3 D

4

Sour

ce

Rs

Ls

ICA:

PWM

PWM

Gen

erat

or

CO

NST

EQU

BL

CO

NST

CP

WM

:=44

IGB

T1

D7

IGB

T2

D8 IG

BT 3

D9

IGB

T 4

D10

IGB

T 5

D5 IG

BT 6

D6

PWM

1

STA

TE1

STA

TE3

STA

TE5

STA

TE2

STA

TE4

STA

TE6

POSI

TTh

Th2

Th01

Qui

ckG

raph

1SY

MP1

.I1A

[A]

SYM

P1.I1

B [A

] SY

MP1

.I1C

[A]

Vdc.

V [V

]

t [s]

0.4k -5

0050

0.1k

0.15

k

0.2k

0.25

k

0.3k

0.35

k 01.

10.

20.

40.

60.

81

Qui

ckG

raph

3SY

MP1

.MI [

N·m

]SY

MP1

.N [r

pm]

t [s]

0.12

k

-20020406080

0.1k

01.

10.

20.

40.

60.

81

MS

3 ~

BA C

SYM

P1

SY

MP

1.N

<PS

/

CPW

M

EQU

FML1

D:=

CP

WM

PE

RIO

D :=

1/2

000

+ V

Vdc

y

tlogi

c:=

RP

ole:

=48

AM

PW

M:=

85

tms:

=(60

/(48*

3))/P

SP

S:=

100

Initi

lizat

io

D

tlogi

c1:=

HE

:=0

HE

:=H

E+1

x

y1

(t-tlo

gic1

)>=t

ms

AN

D (C

PW

M<(

AM

PW

M-2

CP

WM

:=C

PW

M+2

-2*H

E

(t-tlo

gic1

)>=t

ms

AN

D (C

PW

M>=

(AM

PW

M-2

CP

WM

:=A

MP

WM

(t-tlo

gic1

)>=t

ms

AN

D (C

PW

M1<

(AM

PW

M-2

(t-tlo

gic1

)>=t

ms

AN

D (C

PW

M1>

=(A

MP

WM

-2

CP

WM

:=C

PW

M1+

true

true

true

(t-tlo

gic1

)=tm

s A

ND

(CP

WM

>=(A

MP

WM

-2

(t-tlo

gic1

)=tm

s A

ND

(CP

WM

<(A

MP

WM

-2

CP

WM

/85

HE

t [s]

1.

0834

8

CP

WM

:=C

PW

M

tlogi

c1:=

CO

NST

Load

Th.V

AL[

0]>=

2*P

I/

Th.V

AL[

0]>=

4*P

I/Th

.VA

L[0]

>=6*

PI/

Th.V

AL[

0]>=

8*P

I/

Th.V

AL[

0]>=

10*P

ITh

.VA

L[0]

<2*P

I /

true

CP

WM

1:=3

560u

560u

4.35

m0.

086

0.17

6069

5885

1545

k

W+

WM

Sour

ce.V

So

urce

.I [

t[s]

0.2k

-0.2

k0

-0.1

k

0.1k

01.

10.

20.

40.

60.

81

WM

.P [W

]

t[s]

1.2k

-0.2

k

0.2k

0.4k

0.6k

0.8k

01.

10.

20.

40.

60.

81

70,7

%

RM

S

RM

S1

MEA

N V

ALU

E

70,7

%

RM

S

MEA

N1

RM

S 2

RM

S1.V

AL

2.20

898

MEA

N1.

VAL

0.19

3939

k

RM

S2.V

AL

0.12

4511

k

PF

(t-tlo

gic9

)>=0

.2

tlogi

c9:=

tPF

:=(M

EAN

1.VA

L/(R

MS1

.VAL

*RM

S2.V

AL))

true

Pow

er F

acto

r/Effi

cien

cy C

alc.

MA

XIM

UM

MA

X_Is

MAX

_Is.

VAL

R1

:= 1

6.30

9L1

D :=

92.

7274

mL1

Q :=

92.

7274

mKE

:= 0

.223

256

P :=

24

J :=

0.0

33

(SYM

P1.N

*RPo

le*P

I/60)

*0.0

017+

9.5

Fig.

8.2

5. S

tead

y-st

ate

SIM

PLO

RER

® m

odel

of t

he a

gita

tion

spee

d co

ntro

l.

175

D1

D2

D3

D4

Sour

ce

Rs

LsC

1_dc

link

C2_

dclin

k

ICA:

PWM

PWM

Gen

erat

or

CO

NST

EQU

BL

CO

NST

CPW

M:=

3

IGB

T1

D7

IGB

T2

D8 IG

BT3

D9

IGB

T4

D1 0

IGB

T5

D5 IG

BT6

D6

PWM

1

STAT

E1

STAT

E3

STAT

E5

STAT

E2

STAT

E4

STAT

E6

POSI

TTh

Th2

Th01

MS

3 ~

BA C

SYM

P1

(t-tlo

gic)

<RT

AND

Dir=

RT:

=(61

2.5)

/100

PER

IOD

:= 1

/200

+ V

Vdc

y

tlogi

c:=

CO

NST

Load

RPo

le:=

48

AMPW

M:=

tms:

=(60

/(48*

3))/P

PS:=

100

Initi

lizat

ion

tlogi

c1:=

HE:

=0

HE:

=HE+

1

x

y1

(t-tlo

gic1

)>=t

ms

AND

(CPW

M<(

AMPW

MC

PWM

:=C

PWM

+2-2

*H

(t-tlo

gic1

)>=t

ms

AND

(CPW

M>=

(AM

PWM

CPW

M:=

AMPW

(t-tlo

gic1

)>=t

ms

AND

(CPW

M1<

(AM

PWM

(t-tlo

gic1

)>=t

ms

AND

(CPW

M1>

=(AM

PWM

CPW

M:=

CPW

M1+

true

true

true

(t-tlo

gic1

)=tm

s AN

D (C

PWM

>=(A

MPW

M

(t-tlo

gic1

)=tm

s AN

D (C

PWM

<(AM

PWM

CPW

M/8

5

CPW

M:=

CPW

tlogi

c1:=

550u

550u

4.35

m0.

086

AMPL

:= 0

.174

6553

7495

307

Th.V

AL[0

]>=2

*P

Th.V

AL[0

]>=4

*PTh

.VAL

[0]>

=6* P

Th.V

AL[0

]>=8

*P

Th.V

AL[0

]>=1

0*P

Th.V

AL[0

]<2*

P

Load

1

tms:

=tm

s

AMPW

M:=

AMPW

M

(t-tlo

gic)

>=R

IT A

ND

(t-tl

ogic

r)<

AMPW

M<=

MSM

PW

AMPW

M:=

MSM

PW

AMPW

M>M

SMPW

(t-tlo

gicr

)=R

T

RA:

=((S

YMP1

.N-P

S)/P

S)*(

R

(AM

PWM

>=M

SMPW

M) A

ND

(RA

RA:

=0

AMPW

M:=

MSM

PW

(RIT

+RA)

<(2*

176u

RIT

:=2*

176u

AMPW

M:=

MSM

PW

AMPW

M<M

SMPW

RIT

:=R

IT+R

AAM

PWM

:=M

SMPW

true

(RIT

+RA)

<(2*

176u

(RIT

+RA)

>=(2

*176

u

true

RIT

:=R

IT+R

A

(t-tlo

gicr

)>(R

T+PT

) AN

D (P

aT=

CPW

M:=

3tlo

gic2

:=

SYM

P1.N

<=R

tlogi

c2:=

(t-tlo

gic2

)<15

0m

(t-tlo

gic2

)>=1

50m

SYM

P1.N

>RS

RS<

(RS-

4)SY

MP1

.N<=

R

(t-tlo

gicr

)>(R

T+PT

(t-tlo

gicr

)>Pa

TD

ir:=0

PT:=

PT+(

150m

-CT

tlogi

c3:=

RS:

=RS+

4

(t-tlo

gic3

)>10

0mR

S<20

RS:

=20

SYM

P1.N

=H

SYM

P1.N

<>H

RS:

=RS-

4

SYM

P1.N

<>H

tlogi

c4:=

(t-tlo

gic4

)>60

0m

low

ers

onlow

ers

on

?

chan

ge d

ir.

tlogi

c5:=

x:=0

y:=0

z:=0 k:=0

l:=0

m:=

0(t-tlo

gic5

)>25

u

x:=0 m:=

1l:=

1

y:=0

k:=1

z:= 0

AllD

evO

ff

AllD

evO

ff

x:=0

m:=

0l:=

0

y:=0

k:=0

z:=0

tlogi

c2:=

(t-tlo

gic5

)>25

u

l:=1

m:=

1

k:=1

z:= 0

y:=0

x:=0

CPW

M:=

0

PT:=

5000

0/10

0

tmsr

:=(6

0/(4

8*3)

)/(1.

5*P

MSM

PWM

:=8

RIT

:=26

m

RS:

=PS/

PaT:

=0

IF (R

S>35

) R

S:=3

5; E

LSE

IF (R

S<20

) R

S:=2

0; E

LSE

R

IF (P

S<8)

PS:

=

Dir:

=1

true

tlogi

c:=

tlogi

cr:=

(t-tlo

gicr

)=R

T

(t-tlo

gicr

)=R

T

tms:

=(60

/(48*

3))/P

true

tms

RIT

RA AM

PWM

RS

t [s]

0.

7055

17

HE

CPW

M

FML1

D:=

CP

WM

EQU

tlogi

cr

tlogi

c

tlogi

c1

CT:

=(t-t

logi

c2

CPW

M1:

=3

(t-tlo

gic2

)>(3

00/1

000

tlogi

c2:=

true

SYM

P1.N

[rpm

] 0.

1004

12k

LOAD

:= L

oad.

VA

R1

:= 1

6.30

L1D

:= 9

2.72

74m

L1Q

:= 9

2.72

74

KE :=

0.2

232 5

P :=

24

J :=

0.0

3

(SYM

P1.N

*RPo

le*P

I/60)

*0.0

0

Ram

p Ad

just

men

t Com

pans

atio

n M

odel

Ram

p In

crem

ent T

ime

(RIT

) Mod

el(O

nly

in R

ampi

ng)

Coa

st a

nd P

ause

Tim

e M

odel

Plat

eau

Tim

e M

odel

Sing

le P

hase

Rec

tifie

rTh

ree

Phas

e In

vert

er

Fig.

8.2

6. S

IMPL

OR

ER®

mod

el o

f com

mer

cial

ized

ove

rall

agita

tion

spee

d co

ntro

l sys

tem

(tra

nsie

nt+s

tead

y-st

ate)

.

176

Fig. 8.27. Agitation speed control test-bed.

Fig. 8.27 shows the complete experimental setup including pancake motor, hysteresis

dynamometer, power analyzer, dynamometer controller and microcontroller used for

agitation speed control tests.

8.2.3. Simulation Results of the Proposed DTC Method Used in Agitation Cycle

Speed Control

The proposed DTC scheme is simulated to validate its superiority over the

commercialized simple agitation speed control of a laundry washing machine. Several

simulation results are obtained and discussed. These include: real and estimated rotor

position, stator, transient and steady-state speed response under 7 N·m of initial load to

10 N·m of load torque during the acceleration period, stator flux linkage trajectories,

177

amplitude of the stator flux linkage, amplitude of the electromagnetic torque, three phase

motor currents and voltages, and errors in the real and estimated positions.

Although the commercialized simple agitation speed controller simulation is built

and tested in SIMPLORER®, to validate the proposed speed control the complete

proposed DTC scheme has been implemented in MATLAB/SIMULINK®. The reason

for using MATLAB/SIMULINK® is because of the edge of flexibility, speed, superiority

in control and diversity in system tools and components over SIMPLORER®. The

simulations presented in this section are the results of the proposed DTC algorithm as

discussed in Chapter VI. The operating points of the motor speed and load torque are

chosen as 100 rpm steady-state plateau speed, and a 7 N·m to 10 N·m ramp up load

characteristic used during the acceleration period, respectively. The 7 N·m to 10 N·m

load behavior represents the heavy loads for a washing machine. A similar test can be

done for medium and light loads also. One might consider the Light Load range as being

0 N·m-4 N·m, the Medium Load range as 4 N·m-7 N·m, and the Heavy Load range as

7 N·m-10 N·m or up. Depending on the size of the washing machine, these values can be

modified. In the proposed DTC scheme, the ramp up time period is set to 0.35 second,

the plateau region starts from 0.35 second and ends at 1 second and the coast time starts

at 1 second and continues till 1.25 seconds. These values are changeable depending on

the washing profile. You might agree that when the delicate washing cycle is preferred,

then longer ramp up, plateau region and coast times are chosen. For rough washing those

times should be shorter. All parameters are assumed to be known exactly and no

parameter change effect is added to the simulations. The sampling time of the overall

control system is chosen as 25 and the hysteresis bandwidths are 0.005 for torque

control and 0.001 for flux control. Decreasing the sampling time and narrowing the

bandwidths of the hysteresis controllers may mitigate the torque, flux and current ripples

which are common issues of the conventional DTC scheme. The stator flux path will

also become more circular, but the effect of these variations are not included in this

thesis, only a constant sampling time and constant hysteresis bands are assumed in all

simulations.

sμ

178

The conventional DTC scheme is simulated again in MATLAB/SIMULINK® to

compare to the proposed method when the conventional scheme has some discrepancies

in stator flux estimation due to stator resistance variations caused by low speed and/or

high temperature, offset errors in voltage and currents sensing generating drift in the

back-EMF integration because of the wrong initial flux values and/or measurement

errors in voltage and currents.

The stator resistance change effect will be simulated and compared. The

resistance value will be intentionally altered during the ramp up and at steady state to

observe the effect of change on speed and torque response. The amount of deviation

might occur in the stator flux linkage trajectory will also be noted when the change in

resistance affects the speed and torque response. At high load and high temperature, the

resistance may change in the range of 1.3 to 1.75 times of its nominal value. In the

simulations of the conventional DTC of PMSM, the stator resistance value will be

changed to 1.5 times of its nominal value during ramp up and at steady state, especially

under transient loads such as a 7 N·m to 10 N·m incremental load characteristic which is

proportional to speed during the transient state. The drift effect will also be performed by

applying a constant DC value to the measurement of the voltages and currents.

Additionally, the simulation of the initial flux linkage error will be carried out. The

results of those simulations are given in this section and are discussed.

Since the load torque cannot be known initially unless a weight measurement

system is used in the washing machine, starting the motor with a preferred ramp rate

rather than a step speed profile will be a huge turning point in the proposed DTC

scheme. To solve the starting problem, one method which looks at the washing load

capacity chosen by costumer, and estimates the load torque so that smoother starting can

be achieved. The method uses the load information to adjust the initial integral value of

the PI controller. In our tests, load torque (heavy load) value in real washing machine

increases from 7 N·m (assumed initial load on the machine) to 10 N·m during the ramp

up period (load torque is proportional to the speed) and settles to 10 N·m during the

steady state. The plateau speed is selected as 100 rpm. Since there is a constant initial

179

torque, Tem_0, due to ramp up, this can be used in the integrator (PI) as:

Jeq*(100*2*pi/60*P/2)/0.35) if the inertia (Jeq) of the overall system is roughly known,

therefore the final value of the initial integrator value, (Tem_0+8.5), will be the initial

torque, in which 8.5 is an example of the estimated initial load torque by looking at the

customer’s choice of load capacity of the laundry cloths. There will be no position or

speed information from the system start up till the first hall-effect sensor information is

read. From this point till the end of ramp time the estimated speed and the reference

speed will be subtracted from each other as given by:

40ref est− >ω ω rad/s

If the value of this substraction is higher than a predetermined value like 40 rad/s

then at the next stroke the initial integrator value of the PI controller will be decreased

by a reasonable amount. In these tests, 1 N·m is used to be as close to the initial load

torque value (7 N·m is used in the tests) as possible and to minimize the oscillations in

speed and to get more smooth starting. These constant values are based on the ramp up

rate, how fast and smooth the speed profile is desired to be, etc. Since there is no rotor

position or any other sensor information used until the first hall-effect sensor signal the

system starts in open-loop. Regular open-loop starting requires PWM voltage generation

to speed up the motor at a predetermined frequency range or closed-loop starting with

BLDC type of quasi square current control, however, this method does not need a PWM

block to start the motor appropriately without any oscillation. This will eliminate the

disadvantage of using extra code in the proposed control. This is quite beneficial for the

industrial applications such that microcontrollers with less memory can be used.

The simulation results show that the acceleration which is used in the speed

estimation during the complete transient state (ramp up) and in the steady-state degrades

the performance of the speed control. Therefore, the speed estimation information

including the acceleration is obtained only at starting before the rotor position completes

the full cycle. Acceleration due to the ramp rate is never ignored in the speed and

position estimations. For example, when the initial rotor flux linkage is located in the

Sector 1 the following procedures are conducted to estimate the speed:

180

• In Sector 1 (11pi/6-pi/6), in Sector 2 (pi/6-pi/2), and in Sector 3 (pi/2-5pi/6)

acceleration is included in the speed estimation algorithm. This is done to sense

the actual acceleration of the motor more precisely during the transient state to

find a better initial integrator value for PI controller.

• The coast region is simulated without any acceleration in speed which provided a

better speed response than when the acceleration was included to the speed

estimation (acceleration information is still used for position estimation).

At the start up, the rotor flux linkage which might be located in other sectors

rather than the first sector the rotor speed can be obtained in a similar manner such that

the acceleration is included in the speed estimation in only three consecutive sectors. For

the other hall-effect sensor position ranges, the average speed information is preferred

rather than the one with the acceleration and better results are obtained. If a load

disturbance occurs at any point between the two hall-effect sensor signals, the rate of

change of speed will be assumed very wrong looking at the previous acceleration

information for estimating the speed. It has been observed that the proportional

coefficient value should be chosen very low so instability will not occur during the ramp

up. To overcome these problems, the acceleration information is only used in the

estimation at the very beginning of the start up. The acceleration information is required

during transient (ramp up) because the initial ramp rate has to be used in the speed

estimation to determine the speed estimation error when the first few hall-effect sensors

results are read. Once this is known, the initial PI integrator value can be modified

starting from the very next agitation stroke for better starting.

181

8.2.3.1. Transient and Steady-State Response Under Load Condition

The torque characteristics of the PMSM when a 7 N·m to 10 N·m transient load

torque is applied during the 0.35 second ramp up period are shown in Fig. 8.28. In the

same figure the initial integrator value of the PI controller is chosen as 7 N·m with an

additional ramp rate formula. Since there is an initial load on the machine due to the

laundry, the motor should reach the same torque level to start up. If the initial torque

value is small and tries to be incremented for a long time period then desired certain

ramp rate can be deviated so that initial torque value should be as close as load torque

value. As it can be seen from Fig. 8.28 the actual and estimated electromagnetic torque

closely resemble the reference electromagnetic torque generated by PI controller (used

for speed control). After 1 second the plateau region is over and the coast time starts. In

this case the coast time is selected as 0.25 second. It can be observed that the transient

electromagnetic torque during the ramp time is tracking the load torque quite well.

During the steady state, a very pure and constant torque response is achieved. After the

plateau region is over (steady-state), the coast time starts (0.25 second long). During the

coast region, the motor tries to reach zero speed at 0.25 second. Because less attention is

put on the coast time control, the torque response is imperfect but reasonable.

182

0 0.2 0.4 0.6 0.8 1 1.27

8

9

10

Load

Tor

que

(N.m

)

0 0.2 0.4 0.6 0.8 1 1.20

5

10

15

T emre

f (N.m

)

0 0.2 0.4 0.6 0.8 1 1.20

5

10

15

T emes

t (N.m

)

0 0.2 0.4 0.6 0.8 1 1.20

5

10

15

Time (s)

T emac

t (N.m

)

Fig. 8.28. Load torque ( LT ), , , and at a 0.35 second ramp time when

the PI integrator initial value is 7 N·m.


Fig. 8.29 shows the stator flux linkage reference value, sλ∗ , (0.223256 Wb which

is equal to the rotor flux linkage amplitude) estimated stator flux linkage, _s estλ , and

actual stator flux linkage, _s actλ , respectively at a 0.35 second ramp up time when the

initial integrator value of the PI controller is selected as 7 N·m. Three small pikes in the

estimation of the stator flux amplitude are seen in Fig. 8.29 due to small dips that

occurred in the actual torque around 0.2 second. At the end of the plateau time and

starting from the middle of the coast time the actual stator flux amplitude deviates due to

corrupted torque when less attention is put on the speed control during the coast time.

On the other hand, the steady state stator flux linkage values (both estimated and actual)

track the commanded value quite well. The sDλ and sQλ circular trajectory is shown in

Fig. 8.30. The three small spikes occurred in the stator flux linkage estimation can be

also observed in the circular trajectory.

183

0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

Flux

ref (W

b)

0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

Flux

est (W

b)

0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

Time (s)

Flux

actu

al (W

b)

Fig. 8.29. sλ∗ (reference stator flux), _s estλ and _s actλ at a 0.35 second ramp time when


Fig. 8.30. sDλ and sQλ circular trajectory at a 0.35 second ramp time when the PI

integrator initial value is 7 N·m.

184

Almost the same behavior is observed when the initial integrator value of the PI

controller is chosen as 8.5 N·m which is higher than the initial load torque value. The

result of this is that in the beginning of the ramp up speed torque oscillations are

inevitable, as shown in Fig. 8.31. In the same figure one third of the ramp up period

torque drops from 10 N·m (8.5 N·m plus the initial ramp rate causes an almost 2 N·m

initial torque) to around 7 N·m to level out with load torque value.

0 0.2 0.4 0.6 0.8 1 1.27

8

9

10

Load

Tor

que

(N.m

)

0 0.2 0.4 0.6 0.8 1 1.20

5

10

15

T emre

f (N.m

)

0 0.2 0.4 0.6 0.8 1 1.20

5

10

15

T emes

t (N.m

)

0 0.2 0.4 0.6 0.8 1 1.20

5

10

15

Time (s)

T emac

t (N.m

)

Fig. 8.31. Load torque ( LT ), , , and at a 0.35 second ramp time when

the PI integrator initial value is 8.5 N·m.


The stator flux linkage values are represented in Fig. 8.32 when the initial

integrator value of the PI controller is 8.5 N·m. Due to the higher initial torque value

over the load torque in first few milliseconds, the actual stator flux linkage is observed

as higher than what is necessary. It converges to the reference value, however, less than

0.1 second later. That increase in the actual stator flux linkage causes two small pikes in

the estimation of the stator flux linkage as can be seen in Fig. 8.32. After the torque

converges to the reference value, the dips in the actual torque cause the stator flux

185

linkage to reflect the same behavior with two small dips in the ramp. Those two small

dips generate two small pikes in the estimation of the stator flux linkage. Overall the

stator flux linkage results are quite reasonable, especially at steady state. The sDλ and

sQλ circular trajectory is shown in Fig. 8.33. Five small pikes that occurred in stator flux

linkage estimation can be also observed in the circular trajectory.

0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

Flux

ref (W

b)

0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

Flux

est (W

b)

0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

Time (s)

Flux

actu

al (W

b)


the PI integrator initial value is 8.5 N·m.

186

Fig. 8.33. sDλ and sQλ circular trajectory at a 0.35 second ramp time when the PI

integrator initial value is 8.5 N·m.

Figs. 8.34 and 8.35 illustrate the motor’s Phase A, d and q axes currents in the

rotor reference frame when initial integrator of the PI controller is 7 N·m and 8.5 N·m,

respectively. In both cases, the q-axis current in rotor reference frame increases to reflect

the load variation. The d-axis current remains close to zero so that the maximum

torque/current ratio is achieved even in the dynamic response. The q-axis current wave

outlines the wave of the Phase A current which shows that the d- and q-axes currents are

decoupled just like in vector control. As a consequence, even though the DTC scheme

does not directly control the currents (rotor frame d and q axes) when the topology is

investigated in the rotor reference frame similarities with the vector control scheme can

be observed.

The dynamic response of the d- and q-axes currents in the RRF and the Phase A

current under a 7 N·m to 10 N·m transient load during the ramp up when the initial

integrator value of the PI controller is chosen as 7 N·m are shown in Fig. 8.34. Three

hall-effect sensors are used to assist the control algorithm for continuous rotor position

estimation. With the gradual increase of speed, a smooth change in the stator current and

the q-axis current in the RRF are observed during the dynamic state. Fig. 8.34 shows the

187

actual q-axis and d-axis currents in the rotor reference frame obtained by the encoder

and the estimated currents from the position observer method assisted by the three

hall-effect sensors. At steady state, the amplitude of the stator current is essentially

unchanged.

During the steady state, the d-axis current stays close to zero during constant

torque control. The q-axis current varies according to the load (it increases gradually

with the load torque and remains constant when the load torque is constant). All the

figures prove the validity of the proposed DTC method.

0 0.2 0.4 0.6 0.8 1 1.2-2

-1

0

1

2

i as (A

)

0 0.2 0.4 0.6 0.8 1 1.2-1

-0.5

0

0.5

1

1.5

i qrac

t and

i dr

act (A

)

0 0.2 0.4 0.6 0.8 1 1.2-1

0

1

2

Time (s)

i qres

t and

i dr

est (A

)

idrest

iqrest

idract

iqract

Fig. 8.34. Phase-A current ( ), , and when a 0.35 second ramp time is used

when the PI integrator initial value is 7 N·m.

asi dqracti dqresti

188

0 0.2 0.4 0.6 0.8 1 1.2-2

-1

0

1

2

i as (A

)

0 0.2 0.4 0.6 0.8 1 1.2-1

-0.5

0

0.5

1

1.5

i qrac

t and

i dr

act (A

)

0 0.2 0.4 0.6 0.8 1 1.2-1

-0.5

0

0.5

1

1.5

Time (s)

i qres

t and

i dr

est (A

)

idrest

iqrest

idract

iqract


when the PI integrator initial value is 8.5 N·m.

asi dqracti dqresti

The dynamic response of speed when the initial integrator value of the PI

controller is 7 N·m is illustrated in Fig. 8.36. Clearly, the actual speed is capable of

following the reference speed accurately from zero speed to 100 rpm, and also from

100 rpm to zero speed in the coast region. Based on this observation, the estimated rotor

position is displayed in Fig. 8.36. It can be seen that the proposed rotor position

estimation by the help of three cost-effective hall-effect sensors tracks the actual position

quite well.

189

0 0.2 0.4 0.6 0.8 1 1.20

100

200

300

Spe

edes

t and

Spe

edre

f (rad

/sec

)

0 0.2 0.4 0.6 0.8 1 1.2-20

0

20

40

Spe

eder

r

Ref

-Act

(rad

/sec

)

0 0.2 0.4 0.6 0.8 1 1.20

2

4

6

time (sec)

Act

ual a

nd

Est

. Pos

ition

(rad

)

Time (s)

Fig. 8.36. Reference and actual speed, speed error (ref-act), actual and estimated position

when a 0.35 second ramp time is used when the PI integrator initial value is 7 N·m.

In Fig. 8.37, the dynamic response of motor speed is represented when the initial

integrator value of the PI controller is 8.5 N·m. Since the initial torque is higher than the

load torque, more overshoot is observed than when the initial integrator value is 7 N·m.

It takes less than 0.2 second for the speed to converge to the reference value, however,

from now on the actual speed is capable of following the reference speed very well.

Based on this observation, the estimated rotor position is displayed in the same figure. It

can be seen that the proposed rotor position estimation by the help of three cost-effective

hall-effect sensors tracks the actual position quite well.

190

0 0.2 0.4 0.6 0.8 1 1.20

100

200

300

eed es

t and

S

peed

ref (r

ad/s

ec)

0 0.2 0.4 0.6 0.8 1 1.2-40

-20

0

20

40

Spe

eder

r

Ref

-Act

(rad

/sec

)

0 0.2 0.4 0.6 0.8 1 1.20

2

4

6

time (sec)

Act

ual a

nd

Est

. Pos

ition

(rad

)

Time (s)

Fig. 8.37. Reference and actual speed, speed error (ref-act), actual and estimated position

when a 0.35 second ramp time is used when the PI integrator initial value is 8.5 N·m.

The torque characteristics of the PMSM when a 7 N·m. to 10 N·m. transient load

torque is applied during a 0.6125 second ramp up period are shown in Fig. 8.38. In the

same figure the initial integrator value of the PI controller is chosen as 7 N·m with an

additional ramp rate formula. Since there is an initial load on the machine due to the

laundry, the motor should reach the same torque level to start up. As it can be seen from

Fig. 8.39, the actual and estimated electromagnetic torque resemble the reference

electromagnetic torque generated by the PI controller (used for speed control). After the

1.2 seconds plateau region is over the coast time starts. In this case the coast time is

selected as 0.25 second (total stroke time is 1.45 seconds). It can be observed that the

transient electromagnetic torque during ramp time is tracking the reference torque quite

well. During the steady-state a very pure and constant torque response is achieved. After

the plateau region is over (steady-state) the coast time starts (0.25 second long). During

the coast region the motor tries to reach zero speed at 0.25 second.

191

0 0.2 0.4 0.6 0.8 1 1.2 1.40

5

10

15

T emre

f (N.m

)

0 0.2 0.4 0.6 0.8 1 1.2 1.40

5

10

15

T emes

t (N.m

)

0 0.2 0.4 0.6 0.8 1 1.2 1.40

5

10

15

Time (s)

T emac

t (N.m

)

Fig. 8.38. , , and at a 0.6125 second ramp time when the PI integrator

initial value is 7 N·m.


Fig. 8.39 shows the stator flux linkage reference value, sλ∗ , (0.223256 Wb which

is equal to the rotor flux linkage amplitude), the estimated stator flux linkage, _s estλ , and

the actual stator flux linkage, _s actλ , respectively at a 0.6125 second ramp up time when

the initial integrator value of the PI controller is selected as 7 N·m.

For the next simulation the ramp time increases to imitate the delicate washing

cycle. It will give us a good validation opportunity with the conventional agitation

control because most of the time the conventional agitation control is tested under a

0.6125 second ramp up time.

Three small pikes in the estimation of the stator flux amplitude are seen in

Fig. 8.39 due to the small dips in the actual torque at approximately 0.2 second. At the

end of the plateau time and starting from the middle of the coast time the actual stator

flux amplitude deviates due to corrupted torque because less attention is put on the speed

192

control during the coast time. On the other hand, the steady state stator flux linkage

values (both estimated and actual) track the commanded value quite well. The sDλ and

sQλ circular trajectory is shown in Fig. 8.40. Three small pikes that occurred in stator

flux linkage estimation can also be observed in the circular trajectory.

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.1

0.2

0.3

Flux

ref (W

b)

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.1

0.2

0.3

Flux

est (W

b)

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.1

0.2

0.3

Time (s)

Flux

act(W

b)



193

Fig. 8.40. sDλ (X axis) and sQλ (Y axis) circular trajectory at a 0.6125 second ramp time

when the PI integrator initial value is 7 N·m.

The dynamic response of the d and q-axes currents in the RRF and the Phase A

current at a 0.6125 second ramp up time under a 7 N·m to 10 N·m transient load during

the ramp up when the initial integrator value of the PI controller is chosen as 7 N·m are

shown in Fig. 8.41. Three hall-effect sensors assist to the control algorithm for

continuous rotor position estimation. With the gradual increase of speed, a smooth

change in the stator current and q-axis current in the RRF are observed during the

dynamic state. Fig. 8.41 shows the actual q-axis and d-axis currents in the rotor reference

frame obtained by the encoder and the estimated currents from the position observer

method assisted by the three hall-effect sensors. During steady state, the amplitude of the

stator current is essentially unchanged.

194

0 0.2 0.4 0.6 0.8 1 1.2 1.4-2

-1

0

1

2

i as (A

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4-1

-0.5

0

0.5

1

1.5

i qrac

t and

i dr

act (A

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4-0.5

0

0.5

1

1.5

Time (s)

i qres

t and

i dr

est (A

)

idrest

iqrest

idract

iqract

Fig. 8.41. Phase-A current ( ), , and when a 0.6125 second ramp time is

used when the PI integrator initial value is 7 N·m.

asi dqracti dqresti

The dynamic response of the speed at a 0.6125 second ramp up time when the

initial integrator value of the PI controller is 7 N·m is represented in Fig. 8.42. Since the

initial torque is almost as close as load torque minus the overshoot in the speed response

is observed. Based on the observer, the estimated rotor position is displayed in the same

figure. It can be seen that the proposed rotor position estimation with the help of three

cost-effective hall-effect sensors tracks the actual position quite well.

195

0.2 0.4 0.6 0.8 1 1.2 1.40

100

200

300

Spe

edes

t and

Spe

edre

f (rad

/sec

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4-20

0

20

40

Spe

eder

r

Ref

-Act

(rad

/sec

)

0 0.2 0.4 0.6 0.8 1 1.2 1.40

2

4

6

Act

ual a

nd

Est

. Pos

ition

(rad

)

Time (s)

Fig. 8.42. Reference and actual speeds, speed error (ref-act), and actual and estimated

positions when a 0.6125 second ramp time is used when the PI integrator initial value is

7 N·m.

8.2.3.2. Effects of Stator Resistance Variation in Conventional DTC with Encoder

In order to examine the effects of the sR variations of a PM synchronous

machine, a MATLAB/SIMULINK® conventional DTC model was developed. The

DC-link voltage is set at 370 volts. The sR value in the motor model is varied by certain

amount, such as 1.5 times of the nominal value (steady-state, transient and starting etc.).

It should be noted that in all the work related to the resistance change presented here, the

transient load torque is assumed to ramp from 7 N·m to 10 N·m to mimic the real

washing machine load characteristics.

First, a conventional DTC is simulated without the stator resistance change.

Figs. 8.43 and 8.44 show the reference and actual speeds, actual position, and the three

196

phase currents, the reference and estimated electromagnetic torques and the estimated

stator flux linkage when a 0.35 second ramp up time is selected and the stator resistance

is not varied, respectively. Since no resistance change effect is applied and an encoder is

used for speed control, good results are achieved.

0 0.2 0.4 0.6 0.8 1 1.20

100

200

300

Spe

edre

f

an

d S

peed

act (r

ad/s

ec)

0 0.2 0.4 0.6 0.8 1 1.20

2

4

6

Pos

ition

Act

ual (

rad)

Actu

al P

ositi

on (r

ad)

2

0 0.2 0.4 0.6 0.8 1 1.2-2

-1

0

1

Time (s)

Thre

e ph

ase

c

urre

nts

(A)

Fig. 8.43. Reference and actual speeds, actual position, and three phase currents when a

0.35 second ramp time is used (no initial torque, just ramp rate is added). Stator

resistance change effect is not applied.

197

0 0.2 0.4 0.6 0.8 1 1.2 1.40

5

10

15

T emre

f (N.m

)

0 0.2 0.4 0.6 0.8 1 1.2 1.40

5

10

15

T emes

t (N.m

)

0 0.2 0.4 0.6 0.8 1 1.2 1.40.21

0.215

0.22

0.225

0.23

0.235

Time (s)

Flux

est (W

b)

Fig. 8.44. , , and at a 0.35 second ramp time. Stator resistance

change effect is not applied.

emrefT emestT estFlux

A mismatch between the controller set stator resistance and its actual value in the

machine can create instability as shown in Fig. 8.45. This figure shows the simulation

results for a step stator resistance change from 100% to 150% of its nominal value at

0 second and a torque command (7 N·m to 10 N·m) is applied at the start up. The drive

system becomes unstable if controller instrumented stator resistance is lower than its

actual value in the motor. This may be explained as follows: As the motor resistance

increases in the machine, then its current decreases for the same applied voltages, which

decreases the flux and electromagnetic torque. The controller has the opposite effect in

that the decreased currents, which are inputs to the system, cause decreased stator

resistance voltage drops in the calculator resulting in higher flux linkages and

electromagnetic torque estimation. They are compared with their command values

giving larger torque and flux linkages errors resulting in commanding larger voltages

and hence in larger currents leading to a run off condition. The instability result under a

198

step change is questionable as the stator resistance does not change in a step manner in

practice. From the earlier work a linearly decreasing stator resistance provided the same

result. Even for such a gradual change of stator resistance, note that the system becomes

unstable. The controller calculated electromagnetic torque and stator flux linkages are

equal to set values and contrary to the real situation in the machine. Therefore, any

scheme using these values for parameter compensation would not be effective. For

example, the airgap power feedback control for parameter compensation in an indirect

vector controlled drive system is successful but it will not work in the DTC drive

system.

0 0.2 0.4 0.6 0.8 1 1.27

8

9

10

Load

Tor

que

(N.m

)

0 0.2 0.4 0.6 0.8 1 1.2-20

0

20

T emes

t and

T emre

f (N.m

)

0 0.2 0.4 0.6 0.8 1 1.2-50

0

50

T emac

t (N.m

)

0 0.2 0.4 0.6 0.8 1 1.20.2

0.22

0.24

0.26

Flux

est (W

b)

Time (s)

Fig. 8.45. Load torque (TL), , , , and estimated stator flux linkage

under step stator resistance change (1.5x


sR ) at 0 second when ramp time is 0.35 second

when the initial integrator value of the PI is chosen as 7 N·m.

Fig. 8.46 illustrates the actual and reference speeds, actual position, and three

phase motor currents. It can be seen from Fig. 8.45 that due to the lack of control of the

199

torque caused by the stator resistance mismatch, the speed control becomes unnstable

after a few milliseconds of running. If the mismatch in sR is larger, the instability occurs

earlier and vice versa.

0 0.2 0.4 0.6 0.8 1 1.2-200

-100

0

100

200

300

Spe

edre

f

an

d S

peed

act (r

ad/s

ec)

0 0.2 0.4 0.6 0.8 1 1.2

-6

-4

-2

0

2

Pos

ition

Act

ual (

rad)

Actu

al P

ositi

on (r

ad)

5

0 0.2 0.4 0.6 0.8 1 1.2-5

0

Time (s)

Thre

e ph

ase

c

urre

nts

(A)



resistance is increased by 1.5x sR at 0 second (V 370dc = V).

When the stator resistance with a %50 mismatch in sR happens during the

steady-state, as shown in Fig. 8.47, the speed tracks the commanded value very well.

The problem with this, however, is that the actual motor current increases.

200

0 0.2 0.4 0.6 0.8 1 1.20

100

200

300

Spe

edre

f

an

d S

peed

act (r

ad/s

ec)

0 0.2 0.4 0.6 0.8 1 1.20

2

4

6

Pos

ition

Act

ual (

rad)

Actu

al P

ositi

on (r

ad)

0 0.2 0.4 0.6 0.8 1 1.2

-2

-1

0

1

2

Time (s)

Thre

e ph

ase

c

urre

nts

(A)



resistance is increased by 1.5x sR at 0.675 second ( 370dcV = V).

When the stator resistance is increased, the estimated and reference

electromagnetic torque values also increase while the stator flux linkage estimation

remains constant, as depicted in Fig. 8.48.

201

0 0.2 0.4 0.6 0.8 1 1.20

5

10

15

20

T emre

f (N.m

)

0 0.2 0.4 0.6 0.8 1 1.20

5

10

15

20

T emes

t (N.m

)

0 0.2 0.4 0.6 0.8 1 1.20.21

0.215

0.22

0.225

0.23

0.235

Flux

est (W

b)

Time (s)

Fig. 8.48. , , and at a 0.35 second ramp time. Stator resistance is

increased by 1.5x


sR at 0.675 second ( 370dcV = V).

Fig. 8.49 illustrates a good comparison between the estimated electromagnetic

torque, the estimated stator flux linkage value, the actual electromagnetic torque and the

actual stator flux linkage value, respectively. When the resistance increases during

steady-state, the actual value of the electromagnetic torque increases but the actual stator

flux linkage value decreases.

202

0 0.2 0.4 0.6 0.8 1 1.2-5

0

5

10

15

20

T emac

t and

T em

est (N

.m)

0 0.2 0.4 0.6 0.8 1 1.20

0.05

0.1

0.15

0.2

0.25

Flux

act a

nd

Flux

est (W

b)

0 0.2 0.4 0.6 0.8 1 1.2-4

-2

0

2

4

Time (s)

Pos

ition

act a

nd

Pos

ition

est (r

ad)

Temact

Fluxact

Fig. 8.49. , , , , estimated and real rotor positions with a

0.35 second ramp time. Stator resistance is increased by 1.5x

emestT emactT estFlux actFlux

sR at 0.675 second

( 370dcV = V).

Since washing machine agitation speed control needs to work back and forth

between zero and the plateau speed quickly, the resistance change effect will also be

observed in the transient as well as in the start up, even though the change happens in

steady-state. The simulation results show that the change in the stator resistance during

the transient state causes more devastating results in the closed loop speed control, even

with an encoder. Fig. 8.50 shows the electromagnetic torque and the stator flux linkage

behavior of the motor when resistance is incremented by 50% of its nominal value at

0.35/2 second. After a few milliseconds the motor torque becomes unstable which makes

the flux linkage and speed control go unstable. As soon as the step change in the stator

resistance occurs at the transient state the speed goes unstable as shown in Fig. 8.51. As

discussed earlier in this section, in reality the resistance will not change as a step change,

it changes gradually. Recent simulation tests conducted by some authors claim that the

203

gradual change in resistance will only prolong the period of time before instability

occurs.

0 0.2 0.4 0.6 0.8 1 1.2-40

-20

0

20

40

T emac

t and

T em

est (N

.m)

0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

0.4

0.5

Flux

act a

nd

Flux

est (W

b)

0 0.2 0.4 0.6 0.8 1 1.2-4

-2

0

2

4

Pos

ition

act a

nd

Pos

ition

est (r

ad)

Time (s)

Temest

Fluxact

Fluxest

Fig. 8.50. , , , , estimated and real rotor positions with a

0.35 second ramp time. Stator resistance is increased by 1.5x

emestT emactT estFlux actFlux

sR at 0.35/2 second

( 370dcV = V).

204

0 0.05 0.1 0.15 0.2 0.25-50

0

50

100

150

200

Spe

edre

f

an

d S

peed

act (r

ad/s

ec)

0 0.05 0.1 0.15 0.2 0.250

2

4

6

Pos

ition

Act

ual (

rad)

Actu

al P

ositi

on (r

ad)

0 0.05 0.1 0.15 0.2 0.25

-2

-1

0

1

2

Thre

e ph

ase

c

urre

nts

(A)

Time (s)



resistance is increased by 1.5x sR at 0.35/2 second ( 370dcV = V).

205

0 0.05 0.1 0.15 0.2 0.250

5

10

15

20

T emre

f (N.m

)

0 0.05 0.1 0.15 0.2 0.250

5

10

15

20

T emes

t (N.m

)

0 0.05 0.1 0.15 0.2 0.250.21

0.215

0.22

0.225

0.23

0.235

Flux

est (W

b)

Time (s)

Fig. 8.52. , , and with a 0.35 second ramp time. Stator resistance is

increased by 1.5x


sR at 0.35/2 second ( 370dcV = V).

Fig. 8.53 illustrates the estimated stator flux linkage circular trajectory when the

stator resistance is increased by 50% of its nominal value at steady state. The

proportional coefficient of the speed in PI controller is chosen as 9 and the integral

coefficient as 3 for all the simulation that are run.

206

Fig. 8.53. Estimated sDλ (X axis) and sQλ (Y axis) circular trajectory with a 0.35 second

ramp time. Stator resistance is increased by 1.5x sR at 0.675 second (kp=9, ki=3).

The estimated stator flux linkage trajectory when the stator resistance value is

step changed at 0.35/2 second is given in Fig. 8.54. It can clearly be seen that the

resistance value changed in the transient state causes the stator flux linkage locus to be

less smooth than the change occuring at steady-state depicted in Fig. 8.53.

207


ramp time. Stator resistance is increased by 1.5x sR at 0.35/2 second (kp=9, ki=3).

The actual stator flux linkage locus when resistance is varied at the transient state

(0.35/2 second) is shown in Fig. 8.55. Unlike the estimated flux linkage, the actual value

of the stator flux linkage is out of the circular trajectory limit. In this case, the simulation

is stopped at 0.28 second to let the deviation in the flux linkage locus be seen clearly.

208

Fig. 8.55. Actual sDλ (X axis) and sQλ (Y axis) circular trajectory with a 0.35 second

ramp time. Stator resistance is increased by 1.5x sR at 0.35/2 second (stopped at 0.28 s.).

When the stator resistance is changed at steady-state, the circular trajectory

becomes smaller as shown in Fig. 8.56. This occurs when the stator resistance is

increased during the transient state. It has been observed that much quick instability

occurs in the speed control when the stator resistance varies during the transient state

compared to stator resistance varies during steady-state. If the application lets the motor

start once and runs at steady-state then most of the time less instability may be observed

because of the stator resistance change effect. On the other hand, in the washing machine

speed control, the speed has zero, transient, and steady-state conditions such that the

change in stator resistance can occur at any time. This is so that worse case (transient

state) scenario can be considered.

209

Fig. 8.56. Actual sDλ (X axis) and sQλ (Y axis) circular trajectory with a 0.35 second

ramp time. Stator resistance is increased by 1.5x sR at 0.675 second (stopped at 0.28 s.)

8.2.3.3. Drift Problem Occurs in Conventional DTC Scheme

The stator flux is estimated in DTC by integrating the difference between the

input voltage and the voltage drop across the stator resistance. When the stator flux is

indirectly estimated from the integration of the back-EMF, any DC-offset is also

integrated and would eventually lead to large drifts in the stator flux linkage. Operational

amplifiers are normally used to amplify the signal from the current and voltage sensors.

These sensors have a temperature dependent DC-offset. Since the current measurement

path contains many analog devices, DC-offset is an inevitable problem. The error in the

stator flux estimation also causes an error in the torque estimation, so the drive system

may become unstable [35].

The scaling error is another type of current measurement error. Stator currents

are transformed to the voltage signal by current sensors and are transformed into digital

values via low pass filters and A/D converters. The output of a current sensor must be

210

scaled to fit the input range of an A/D converter to get the real value of a current. During

this process, some scaling error may be introduced. Obviously, drift compensation is an

important factor in a practical implementation of the integration, since drift can cause a

large error in the estimated flux linkage. The error in the estimated flux linkage can

cause error in the estimated torque and speed [35].

Figs. 8.57 through 8.59 show the experimental results when a 0.1 A (~ 4% of the

rated current) offset is introduced. Figure 8.57 shows the reference and actual speeds and

the actual position and three phase motor currents with an offset added. It is seen that the

speed quickly goes unstable at start up and the accumulation of the offset error causes

the estimated stator flux to drift from its origin as shown in Fig. 8.59. Fig. 8.58 shows

the load torque, the reference and actual electromagnetic torque and also the reference

and actual stator flux linkages. Starting at 0.4 second, the stator flux linkage oscillates

too much and its amplitude increases greatly and from around 0.5 second the actual

developed torque oscillates and in a short time goes unstable (not shown in the figure).

211

0 0.1 0.2 0.3 0.4 0.5 0.6-3000

-2000

-1000

0

1000

Spe

edre

f

an

d S

peed

act (r

ad/s

)

0 0.1 0.2 0.3 0.4 0.5 0.6-10

-5

0

5

Pos

ition

Act

ual (

rad)

0 0.1 0.2 0.3 0.4 0.5 0.6-10

-5

0

5

10

Time (s)

Thre

e ph

ase

c

urre

nts

(A)

Speedref (rad/s)


0.35 second ramp time is used (no initial torque, just ramp rate is added). A 0.1 A

current offset is added to the D- and Q-axes currents in the SRF ( V). 370dcV =

Actu

al P

ositi

on (r

ad)

212

0 0.1 0.2 0.3 0.4 0.5 0.67

8

9

10

Load

Tor

que

(N.m

)

0 0.1 0.2 0.3 0.4 0.5 0.6-50

0

50

T emre

f (N.m

)

0 0.1 0.2 0.3 0.4 0.5 0.6-50

0

50

T emac

t (N.m

)

0 0.1 0.2 0.3 0.4 0.5 0.60.2

0.4

0.6

0.8

Time (s)

Flux

act a

nd

Flux

est (W

b) Fluxact

Fig. 8.58. Load torque (TL), , , , and actual stator flux linkage.

A 0.1 A offset occurs on the D- and Q-axes currents in the SRF when the ramp time is

0.35 second.

emrefT emactT estFlux

213


ramp time. A 0.1 A offset is included in the D- and Q-axes currents in the SRF

( 37dcV 0= V). (kp=9, ki=3).

8.2.3.4. Overview of the Offset Error Occurred in the Proposed DTC Scheme

Unlike the conventional DTC scheme, the test results shown below prove that

around a 4% offset in the d- and q-axes currents in the RRF does not affect the

closed-loop speed control of the proposed DTC scheme. When there is an initial 7 N·m

load torque in the washing machine which gradually increases to a 10 N·m steady-state

limit proportional to the speed (choosing 7.5 N·m. initial integrator value in the speed PI

controller) results in smooth starting in the case of an offset added to the currents. If a

7 N·m initial value is selected, a lower smoothness in the speed response is observed.

Since the adaptive initial value correction algorithm (discussed earlier in this section) is

used, after a few strokes the right value would be reached even if there is an offset in the

measurement systems.

214

Figs. 8.60 to 8.64 display several simulation results when a 0.1 A offset error is included

to the d- and q-axes currents in the RRF. Similar behaviors are observed in the results

without the offset error discussed earlier in this section.

0 0.2 0.4 0.6 0.8 1 1.20

100

200

300

Spe

edes

t and

Spe

edre

f (rad

/s)

0 0.2 0.4 0.6 0.8 1 1.2-20

0

20

40

Spe

eder

r

Ref

-Act

(rad

/s)

0 0.2 0.4 0.6 0.8 1 1.20

2

4

6

Act

ual a

nd

Est

. Pos

ition

(rad

)

Time (s)

Fig. 8.60. Reference and actual speeds, speed error (ref-act), estimated and actual

positions when a 0.35 second ramp time is used (7.5 N·m initial torque is used in the PI’s

integrator with appropriate ramp rate). A 0.1 A offset is included in the d- and q-axes

currents in the RRF ( 370dcV = V).

215

0 0.2 0.4 0.6 0.8 1 1.2-2

-1

0

1

2

i as (A

)

0 0.2 0.4 0.6 0.8 1 1.2-2

-1

0

1

2

i qrac

t and

i dr

act (A

)

0 0.2 0.4 0.6 0.8 1 1.2-1

0

1

2

i qres

t and

i dr

est (A

)

Time (s)

idract

iqract

idrest

iqrest


with a PI integrator initial value of 7.5 N·m. A 0.1 A offset is included in the d- and

q-axes current in the RRF (

asi dqracti dqresti

370dcV = V).

216

0 0.2 0.4 0.6 0.8 1 1.27

8

9

10

Load

Tor

que

(N.m

)

0 0.2 0.4 0.6 0.8 1 1.20

5

10

15

T emre

f (N.m

)

0 0.2 0.4 0.6 0.8 1 1.20

5

10

15

T emes

t (N.m

)

0 0.2 0.4 0.6 0.8 1 1.20

5

10

15

T emac

t (N.m

)

Time (s)

Fig. 8.62. Load torque (TL), , , and . A 0.1 A offset occurs on d- and

q-axes currents in the RRF when ramp time is 0.35 second ( V).


370dcV =

217

0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

Flux

ref (W

b)

0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

Flux

est (W

b)

0 0.2 0.4 0.6 0.8 1 1.20.1

0.15

0.2

0.25

0.3

0.35

Time (s)

Flux

actu

al (W

b)

Fig. 8.63. , , when a 0.35 second ramp time is used with a PI

integrator initial value of 7.5 N·m. A 0.1 A offset is included to the d- and q-axes

currents in the RRF (


370dcV = V).

218

Fig. 8.64. Estimated sDλ (X axis) and sQλ (Y axis) circular trajectory at a 0.35 second

ramp time with a PI integrator initial value of 7.5 N·m. A 0.1 A offset is included in the

d- and q-axes currents in the RRF ( 370dcV = V). (kp=9, ki=3).

8.2.3.5. Overview of the Rotor Flux Linkage Error Occurred in the Proposed DTC

Scheme

Similar to conventional the DTC scheme having sR variations due to temperature

change and low the speed effect, in the proposed scheme the rotor flux linkage value is

considered as constant in most of the cases. The exception to this is in high temperature

conditions. Some experiments show that amplitude of the rotor flux linkage can vary up

to 20-30 percent of its nominal value at around 100 C° on the motor stator windings. The

value of the rotor flux linkage amplitude decreases when temperature rises. Several

simulations are conducted to examine the effect of the rotor flux linkage change effect

when the motor heats up on the closed-loop control of the proposed DTC scheme. The

rotor flux linkage amplitude is step changed when the motor speeds up (at

0.35/2 second) proportional to the load torque. This is considered the worse case

219

situation. The results, however, prove that the 30% decrease in the rotor flux linkage

amplitude does not have a large impact on the closed-loop speed control.

Some of the simulation results when the rotor flux linkage value is changed by

70% of its nominal value are given from Figs. 8.65 through 8.69.

0 0.2 0.4 0.6 0.8 1 1.20

100

200

300

Spe

edes

t and

Spe

edre

f (rad

/s)

0 0.2 0.4 0.6 0.8 1 1.2-20

0

20

40

Spe

eder

r

Ref

-Act

(rad

/s)

0 0.2 0.4 0.6 0.8 1 1.20

2

4

6

Act

ual a

nd

Est

. Pos

ition

(rad

)

Time (s)

Fig. 8.65. Reference and actual speeds, speed error (ref-act), estimated and actual

positions when a 0.35 second ramp time is used (7 N·m. initial torque is used in the PI’s

integrator with appropriate ramp rate). 0.7x is used at 0.35/2 second in the motor

model (

rλ

370dcV = V). (kp=9, ki=3).

220

0 0.2 0.4 0.6 0.8 1 1.2-2

-1

0

1

2

i as (A

)

0 0.2 0.4 0.6 0.8 1 1.2-1

-0.5

0

0.5

1

1.5

i qrac

t and

i dr

act (A

)

0 0.2 0.4 0.6 0.8 1 1.2-1

0

1

2

i qres

t and

i dr

est (A

)

Time (s)


with a PI integrator initial value of 7 N·m. 0.7x is used at 0.35/2 second in the motor

model (

asi dqracti dqresti

rλ

370dcV = V). (kp=9, ki=3).

221

0 0.2 0.4 0.6 0.8 1 1.27

8

9

10

Load

Tor

que

(N.m

)

0 0.2 0.4 0.6 0.8 1 1.20

5

10

15

T emre

f (N.m

)

0 0.2 0.4 0.6 0.8 1 1.20

5

10

15

T emes

t (N.m

)

0 0.2 0.4 0.6 0.8 1 1.20

5

10

15

T emac

t (N.m

)

Time (s)

Fig. 8.67. Load torque (TL), , , and . 0.7x is used at 0.35/2 second in

the motor model (

emrefT emestT emactT rλ

370dcV = V). (kp=9, ki=3).

222

0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

Flux

ref (W

b)

0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

Flux

est (W

b)

0 0.2 0.4 0.6 0.8 1 1.20.1

0.15

0.2

0.25

0.3

0.35

Time (s)

Flux

actu

al (W

b)

Fig. 8.68. , , when a 0.35 second ramp time is used with a PI

integrator initial value of 7 N·m. 0.7x is used at 0.35/2 second in the motor model

( 3


rλ

7dcV 0= V). (kp=9, ki=3).

223

Fig. 8.69. Estimated sDλ (X axis) and sQλ (Y axis) circular trajectory at a 0.35 second

ramp time with a PI integrator initial value of 7 N·m. 0.7x is used at 0.35/2 second in

the motor model (

rλ

370dcV = V). (kp=9, ki=3).

224

CHAPTER IX

SUMMARY AND FUTURE WORK

9.1. Summary of the Work and Conclusion

The primary contribution of this thesis work is the development, analysis and

simulation verification of DTC of PMSM algorithm using three hall-effect sensors that is

based on rotor reference frame topology for the improvement of the commercial

top-loaded laundry washing machine agitation speed control system.

Since DTC scheme is often categorized as a sensorless torque control technique

because stator flux linkage and electromagnetic torque can easily be estimated without a

mechanical position or speed sensor at all but very low speeds. At very low and zero

speeds, the technique suffers the same performance degradation as any sensorless

method that is based on back-EMF integration. At very low speeds the back-EMF

information is very weak and quietly comparable with the supply voltage in addition of

resistance variation in conjunction with current and voltage sensing errors (offset errors

causing a drift in integration). On the other hand, since proposed method does not use

any integration (back-EMF integration), and uses the rotor position information which is

updated at every 60 electrical degrees by the help of three cost-effective hall-effect

sensors very low speed control is performed without any problem unlike basic DTC

scheme.

Observer based type of position estimation techniques have problem of

accumulating the position errors. Hall-effect sensor correction method gives a solution

for these problems. By correcting the estimated position every cycle, no compensation

algorithm is required and accumulating error is also eliminated.

225

This method eliminates the need of back-EMF integration to estimate the stator

flux linkage; instead, rotor reference frame solution which relies only on the parameters

of rotor flux linkage amplitude and dq-axis synchronous inductances rather than more

sensitive stator resistance used in basic DTC scheme, is proposed for the same purpose.

First, the mathematical basis of the developed method is derived and explained in detail

then simulation model is built in MATLAB/SIMULINK® to verify the proposed method

both in transient and steady-state condition with transient load starting from 7 N·m at

start up and increase up to 10 N·m proportional to speed and the results are given and are

discussed. As it can be seen in Section 8.2.3 of Chapter VIII, results of the simulations

verify the feasibility of the proposed control scheme over the commercialized one. One

conclusion can be drawn from the results is that the proposed method shows superior

speed and torque responds when compare with the commercial agitate speed control

system. Comparing to conventional DTC scheme when it has resistance variations and

offset error causing drift in the stator flux linkage and eventually the instability in

closed-loop speed control, those effects are included to the proposed DTC scheme also

but no speed or torque instability are noted. Instead of resistance variation, rotor flux

linkage amplitude is varied in the proposed DTC topology because only two motor

parameters (rotor flux linkage amplitude and synchronous inductance) are used in the

scheme.

9.2. Future Work

More work could be done in the future to improve the DTC of PMSM drive in

both quality and economical manner. Some of those possible works are listed below:

• Current sensorless technique can be considered.

• Parameter change effect and their compensation can be included.

226

• Current model used in this thesis can be combined with voltage model to

improve the performance of the speed control.

• One of the efficient ways of sensorless control techniques, high frequency

injection, can be implemented.

• One and/or two hall-effect sensors instead of three can be used.

• Saturation effect on the synchronous inductance can be included.

• Temperature arise impacting on rotor flux linkage amplitude can be

studied.

227

REFERENCES

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232

APPENDIX A

Fig. A-1. Specifications of the Fujitsu MB907462A microcontroller.

233

Fig A-1. (continued).

234

Fig. A-2. Theoretical background of active power, apparent power, power factor and

connection of the power analyzer to the load (motor).

235

D

TC o

f IPM

SM w

ith 3

Hal

l-Effe

ct S

enso

rs

Lam

bda_

s*

0-1-

2-3-

4-5-

6-7

Ram

p_sp

eed_

ref

f(u)

rem

func

.

0

lam

bda_

s*

1 s 1 s

Zero

-Ord

erHo

ld fo

r wre

[D]

Wre

f_el

ec

Sw_k

s

Vas

Vbs

Vcs

Volt

Sour

ce

Vol.

Vect

or M

atrix

trq sec

phi

vect

or

Vect

or S

elec

tion

Torq

uePl

ot

Torq

uehy

stere

sisco

ntro

ller

0

Tem

_estTL

TL_S

tep_

up_d

own

Switc

h

Info

rho

sect

or1

Sect

or F

inde

r

100*

2*pi

/60*

24

Ref_

elec

_ste

ady-

state

_spe

ed

Ram

p

MAT

LAB

Func

tion

Prop

osed

Alg

orith

m

0

Posit

ion(

0-2p

i)

0

Posit

ion

du/d

t

Pos-t

o-wr

e

err.

Te*

PI c

ontro

ller

Mot

or_C

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nts

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_ele

c

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bda_

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da_s

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Plot

Flux

hyste

resis

cont

rolle

r

0

Est_

elec

_spe

ed

Est_

elec

_pos

(0-2

pi)

Elec

_spe

eds

em

Dec

-Bin

Bin-

Vec

Dec2

Vec

fl_s_

0

Cloc

k2

0

Cloc

k

0

Actu

al_w

re

Actu

al_m

ech_

spee

d

Actu

al_e

lec_

pos (

0-2p

i) Ac

tual

_ele

c_po

s

0

Wm

ech

Tem

*

Fig.

A-3

. MA

TLA

B/S

IMU

LIN

K®

mod

el o

f the

pro

pose

d D

TC.

236

MOTOR PARAMETERS

p (number of poles) : 48

ratedV : 325 V

peakI : 3.5 A

2r kλ = e (rotor flux linkage): 0.223256 Vs/rad

sR : 16.30983 Ω

sL : 61.8183 mH

d qL L= : 92.72745 mH

peakT : 30 N·m

237

VITA

Salih Baris Ozturk received his Bachelor of Science degree from Istanbul

Technical University, Istanbul, Turkey, in June 2000. He worked in the Electrical

Machines and Power Electronics Laboratory, in the Department of Electrical

Engineering at Texas A&M University, College Station, Texas, with Dr. Hamid A.

Toliyat as his advisor, and he obtained a M.S. degree in December 2005. He is currently

pursuing a Ph.D. degree in the same university.

Salih Baris Ozturk is a student member of the IEEE.

He can be reached in care of:

Dr. Hamid A. Toliyat Electrical Machines & Power Electronics Laboratory Department of Electrical Engineering TAMU 3128 Texas A&M University College Station, Texas 77843-3128

Documents

Modelling, simulation and analysis of low-cost direct torque control of PMSM using hall-effect se